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Theorem List for Metamath Proof Explorer - 19101-19200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremchpdmatlem1 19101 Lemma 1 for chpdmat 19104. (Contributed by AV, 18-Aug-2019.)
 |-  C  =  ( N CharPlyMat  R )   &    |-  P  =  (Poly1 `  R )   &    |-  A  =  ( N Mat  R )   &    |-  S  =  (algSc `  P )   &    |-  B  =  ( Base `  A )   &    |-  X  =  (var1 `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  G  =  (mulGrp `  P )   &    |-  .-  =  ( -g `  P )   &    |-  Q  =  ( N Mat  P )   &    |-  .1.  =  ( 1r `  Q )   &    |- 
 .x.  =  ( .s `  Q )   &    |-  Z  =  (
 -g `  Q )   &    |-  T  =  ( N matToPolyMat  R )   =>    |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B ) 
 ->  ( ( X  .x.  .1.  ) Z ( T `
  M ) )  e.  ( Base `  Q ) )
 
Theoremchpdmatlem2 19102 Lemma 2 for chpdmat 19104. (Contributed by AV, 18-Aug-2019.)
 |-  C  =  ( N CharPlyMat  R )   &    |-  P  =  (Poly1 `  R )   &    |-  A  =  ( N Mat  R )   &    |-  S  =  (algSc `  P )   &    |-  B  =  ( Base `  A )   &    |-  X  =  (var1 `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  G  =  (mulGrp `  P )   &    |-  .-  =  ( -g `  P )   &    |-  Q  =  ( N Mat  P )   &    |-  .1.  =  ( 1r `  Q )   &    |- 
 .x.  =  ( .s `  Q )   &    |-  Z  =  (
 -g `  Q )   &    |-  T  =  ( N matToPolyMat  R )   =>    |-  ( ( ( ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B ) 
 /\  i  e.  N )  /\  j  e.  N )  /\  i  =/=  j
 )  /\  ( i M j )  =  .0.  )  ->  (
 i ( ( X 
 .x.  .1.  ) Z ( T `  M ) ) j )  =  ( 0g `  P ) )
 
Theoremchpdmatlem3 19103 Lemma 3 for chpdmat 19104. (Contributed by AV, 18-Aug-2019.)
 |-  C  =  ( N CharPlyMat  R )   &    |-  P  =  (Poly1 `  R )   &    |-  A  =  ( N Mat  R )   &    |-  S  =  (algSc `  P )   &    |-  B  =  ( Base `  A )   &    |-  X  =  (var1 `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  G  =  (mulGrp `  P )   &    |-  .-  =  ( -g `  P )   &    |-  Q  =  ( N Mat  P )   &    |-  .1.  =  ( 1r `  Q )   &    |- 
 .x.  =  ( .s `  Q )   &    |-  Z  =  (
 -g `  Q )   &    |-  T  =  ( N matToPolyMat  R )   =>    |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B )  /\  K  e.  N )  ->  ( K ( ( X 
 .x.  .1.  ) Z ( T `  M ) ) K )  =  ( X  .-  ( S `  ( K M K ) ) ) )
 
Theoremchpdmat 19104* The characteristic polynomial of a diagonal matrix. (Contributed by AV, 18-Aug-2019.) (Proof shortened by AV, 21-Nov-2019.)
 |-  C  =  ( N CharPlyMat  R )   &    |-  P  =  (Poly1 `  R )   &    |-  A  =  ( N Mat  R )   &    |-  S  =  (algSc `  P )   &    |-  B  =  ( Base `  A )   &    |-  X  =  (var1 `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  G  =  (mulGrp `  P )   &    |-  .-  =  ( -g `  P )   =>    |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B ) 
 /\  A. i  e.  N  A. j  e.  N  ( i  =/=  j  ->  ( i M j )  =  .0.  )
 )  ->  ( C `  M )  =  ( G  gsumg  ( k  e.  N  |->  ( X  .-  ( S `
  ( k M k ) ) ) ) ) )
 
Theoremchpscmat 19105* The characteristic polynomial of a (nonempty!) scalar matrix. (Contributed by AV, 21-Aug-2019.)
 |-  C  =  ( N CharPlyMat  R )   &    |-  P  =  (Poly1 `  R )   &    |-  A  =  ( N Mat  R )   &    |-  X  =  (var1 `  R )   &    |-  G  =  (mulGrp `  P )   &    |-  .^  =  (.g `  G )   &    |-  D  =  { m  e.  ( Base `  A )  |  E. c  e.  ( Base `  R ) A. i  e.  N  A. j  e.  N  ( i m j )  =  if ( i  =  j ,  c ,  ( 0g
 `  R ) ) }   &    |-  S  =  (algSc `  P )   &    |-  .-  =  ( -g `  P )   =>    |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( M  e.  D  /\  I  e.  N  /\  A. n  e.  N  ( n M n )  =  E ) ) 
 ->  ( C `  M )  =  ( ( # `
  N )  .^  ( X  .-  ( S `
  E ) ) ) )
 
Theoremchpscmat0 19106* The characteristic polynomial of a (nonempty!) scalar matrix, expressed with its diagonal element. (Contributed by AV, 21-Aug-2019.)
 |-  C  =  ( N CharPlyMat  R )   &    |-  P  =  (Poly1 `  R )   &    |-  A  =  ( N Mat  R )   &    |-  X  =  (var1 `  R )   &    |-  G  =  (mulGrp `  P )   &    |-  .^  =  (.g `  G )   &    |-  D  =  { m  e.  ( Base `  A )  |  E. c  e.  ( Base `  R ) A. i  e.  N  A. j  e.  N  ( i m j )  =  if ( i  =  j ,  c ,  ( 0g
 `  R ) ) }   &    |-  S  =  (algSc `  P )   &    |-  .-  =  ( -g `  P )   =>    |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( M  e.  D  /\  I  e.  N  /\  A. n  e.  N  ( n M n )  =  ( I M I ) ) ) 
 ->  ( C `  M )  =  ( ( # `
  N )  .^  ( X  .-  ( S `
  ( I M I ) ) ) ) )
 
Theoremchpscmatgsumbin 19107* The characteristic polynomial of a (nonempty!) scalar matrix, expressed as finite group sum of binomials. (Contributed by AV, 2-Sep-2019.)
 |-  C  =  ( N CharPlyMat  R )   &    |-  P  =  (Poly1 `  R )   &    |-  A  =  ( N Mat  R )   &    |-  X  =  (var1 `  R )   &    |-  G  =  (mulGrp `  P )   &    |-  .^  =  (.g `  G )   &    |-  D  =  { m  e.  ( Base `  A )  |  E. c  e.  ( Base `  R ) A. i  e.  N  A. j  e.  N  ( i m j )  =  if ( i  =  j ,  c ,  ( 0g
 `  R ) ) }   &    |-  S  =  (algSc `  P )   &    |-  .-  =  ( -g `  P )   &    |-  F  =  (.g `  P )   &    |-  H  =  (mulGrp `  R )   &    |-  E  =  (.g `  H )   &    |-  I  =  ( invg `  R )   &    |-  .x.  =  ( .s `  P )   =>    |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( M  e.  D  /\  J  e.  N  /\  A. n  e.  N  ( n M n )  =  ( J M J ) ) ) 
 ->  ( C `  M )  =  ( P  gsumg  (
 l  e.  ( 0
 ... ( # `  N ) )  |->  ( ( ( # `  N )  _C  l ) F ( ( ( ( # `  N )  -  l ) E ( I `  ( J M J ) ) )  .x.  ( l  .^  X ) ) ) ) ) )
 
Theoremchpscmatgsummon 19108* The characteristic polynomial of a (nonempty!) scalar matrix, expressed as finite group sum of scaled monomials. (Contributed by AV, 2-Sep-2019.)
 |-  C  =  ( N CharPlyMat  R )   &    |-  P  =  (Poly1 `  R )   &    |-  A  =  ( N Mat  R )   &    |-  X  =  (var1 `  R )   &    |-  G  =  (mulGrp `  P )   &    |-  .^  =  (.g `  G )   &    |-  D  =  { m  e.  ( Base `  A )  |  E. c  e.  ( Base `  R ) A. i  e.  N  A. j  e.  N  ( i m j )  =  if ( i  =  j ,  c ,  ( 0g
 `  R ) ) }   &    |-  S  =  (algSc `  P )   &    |-  .-  =  ( -g `  P )   &    |-  F  =  (.g `  P )   &    |-  H  =  (mulGrp `  R )   &    |-  E  =  (.g `  H )   &    |-  I  =  ( invg `  R )   &    |-  .x.  =  ( .s `  P )   &    |-  Z  =  (.g `  R )   =>    |-  ( ( ( N  e.  Fin  /\  R  e.  CRing )  /\  ( M  e.  D  /\  J  e.  N  /\  A. n  e.  N  ( n M n )  =  ( J M J ) ) ) 
 ->  ( C `  M )  =  ( P  gsumg  (
 l  e.  ( 0
 ... ( # `  N ) )  |->  ( ( ( ( # `  N )  _C  l ) Z ( ( ( # `  N )  -  l
 ) E ( I `
  ( J M J ) ) ) )  .x.  ( l  .^  X ) ) ) ) )
 
Theoremchp0mat 19109 The characteristic polynomial of the zero matrix. (Contributed by AV, 18-Aug-2019.)
 |-  C  =  ( N CharPlyMat  R )   &    |-  P  =  (Poly1 `  R )   &    |-  A  =  ( N Mat  R )   &    |-  X  =  (var1 `  R )   &    |-  G  =  (mulGrp `  P )   &    |-  .^  =  (.g `  G )   &    |-  .0.  =  ( 0g `  A )   =>    |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  ( C `  .0.  )  =  ( ( # `  N )  .^  X ) )
 
Theoremchpidmat 19110 The characteristic polynomial of the identity matrix. (Contributed by AV, 19-Aug-2019.)
 |-  C  =  ( N CharPlyMat  R )   &    |-  P  =  (Poly1 `  R )   &    |-  A  =  ( N Mat  R )   &    |-  X  =  (var1 `  R )   &    |-  G  =  (mulGrp `  P )   &    |-  .^  =  (.g `  G )   &    |-  I  =  ( 1r `  A )   &    |-  S  =  (algSc `  P )   &    |- 
 .1.  =  ( 1r `  R )   &    |-  .-  =  ( -g `  P )   =>    |-  ( ( N  e.  Fin  /\  R  e.  CRing
 )  ->  ( C `  I )  =  ( ( # `  N )  .^  ( X  .-  ( S `  .1.  )
 ) ) )
 
Theoremchmaidscmat 19111 The characteristic polynomial of a matrix multiplied with the identity matrix is a scalar matrix. (Contributed by AV, 30-Oct-2019.) (Revised by AV, 19-Dec-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  C  =  ( N CharPlyMat  R )   &    |-  P  =  (Poly1 `  R )   &    |-  E  =  ( Base `  P )   &    |-  Y  =  ( N Mat  P )   &    |-  K  =  ( Base `  Y )   &    |-  .x.  =  ( .s `  Y )   &    |-  .0.  =  ( 0g `  P )   &    |- 
 .1.  =  ( 1r `  Y )   &    |-  S  =  ( N ScMat  P )   =>    |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B ) 
 ->  ( ( C `  M )  .x.  .1.  )  e.  S )
 
11.5.2  The characteristic factor function G

In this subsection the function  G  =  ( n  e.  NN0  |->  if ( n  =  0 ,  (  .0.  .-  (
( T `  M
)  .X.  ( T `  ( b `  0
) ) ) ) ,  if ( n  =  ( s  +  1 ) ,  ( T `  ( b `
 s ) ) ,  if ( ( s  +  1 )  <  n ,  .0.  ,  ( ( T `  ( b `  (
n  -  1 ) ) )  .-  (
( T `  M
)  .X.  ( T `  ( b `  n
) ) ) ) ) ) ) ) is discussed. This function is involved in the representation of the product of the characteristic matrix of a given matrix and its adjunct as an infinite sum, see cpmadugsum 19141. Therefore, this function is called "characteristic factor function" (in short "chfacf") in the following. It plays an important role in the proof of the Cayley-Hamilton theorem, see cayhamlem1 19129, cayhamlem3 19150 and cayhamlem4 19151.

 
Theoremfvmptnn04if 19112* The function values of a mapping from the nonnegative integers with four distinct cases. (Contributed by AV, 10-Nov-2019.)
 |-  G  =  ( n  e.  NN0  |->  if ( n  =  0 ,  A ,  if ( n  =  S ,  C ,  if ( S  <  n ,  D ,  B ) ) ) )   &    |-  ( ph  ->  S  e.  NN )   &    |-  ( ph  ->  N  e.  NN0 )   &    |-  ( ph  ->  Y  e.  V )   &    |-  ( ( ph  /\  N  =  0 ) 
 ->  Y  =  [_ N  /  n ]_ A )   &    |-  ( ( ph  /\  0  <  N  /\  N  <  S )  ->  Y  =  [_ N  /  n ]_ B )   &    |-  ( ( ph  /\  N  =  S ) 
 ->  Y  =  [_ N  /  n ]_ C )   &    |-  ( ( ph  /\  S  <  N )  ->  Y  =  [_ N  /  n ]_ D )   =>    |-  ( ph  ->  ( G `  N )  =  Y )
 
Theoremfvmptnn04ifa 19113* The function value of a mapping from the nonnegative integers with four distinct cases for the first case. (Contributed by AV, 10-Nov-2019.)
 |-  G  =  ( n  e.  NN0  |->  if ( n  =  0 ,  A ,  if ( n  =  S ,  C ,  if ( S  <  n ,  D ,  B ) ) ) )   &    |-  ( ph  ->  S  e.  NN )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ( ph  /\  N  =  0  /\  [_ N  /  n ]_ A  e.  V )  ->  ( G `
  N )  = 
 [_ N  /  n ]_ A )
 
Theoremfvmptnn04ifb 19114* The function value of a mapping from the nonnegative integers with four distinct cases for the second case. (Contributed by AV, 10-Nov-2019.)
 |-  G  =  ( n  e.  NN0  |->  if ( n  =  0 ,  A ,  if ( n  =  S ,  C ,  if ( S  <  n ,  D ,  B ) ) ) )   &    |-  ( ph  ->  S  e.  NN )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ( ph  /\  (
 0  <  N  /\  N  <  S )  /\  [_ N  /  n ]_ B  e.  V )  ->  ( G `  N )  =  [_ N  /  n ]_ B )
 
Theoremfvmptnn04ifc 19115* The function value of a mapping from the nonnegative integers with four distinct cases for the third case. (Contributed by AV, 10-Nov-2019.)
 |-  G  =  ( n  e.  NN0  |->  if ( n  =  0 ,  A ,  if ( n  =  S ,  C ,  if ( S  <  n ,  D ,  B ) ) ) )   &    |-  ( ph  ->  S  e.  NN )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ( ph  /\  N  =  S  /\  [_ N  /  n ]_ C  e.  V )  ->  ( G `
  N )  = 
 [_ N  /  n ]_ C )
 
Theoremfvmptnn04ifd 19116* The function value of a mapping from the nonnegative integers with four distinct cases for the forth case. (Contributed by AV, 10-Nov-2019.)
 |-  G  =  ( n  e.  NN0  |->  if ( n  =  0 ,  A ,  if ( n  =  S ,  C ,  if ( S  <  n ,  D ,  B ) ) ) )   &    |-  ( ph  ->  S  e.  NN )   &    |-  ( ph  ->  N  e.  NN0 )   =>    |-  ( ( ph  /\  S  <  N  /\  [_ N  /  n ]_ D  e.  V )  ->  ( G `
  N )  = 
 [_ N  /  n ]_ D )
 
Theoremchfacfisf 19117* The "characteristic factor function" is a function from the nonnegative integers to polynomial matrices. (Contributed by AV, 8-Nov-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  P  =  (Poly1 `  R )   &    |-  Y  =  ( N Mat  P )   &    |-  .X. 
 =  ( .r `  Y )   &    |-  .-  =  ( -g `  Y )   &    |-  .0.  =  ( 0g `  Y )   &    |-  T  =  ( N matToPolyMat  R )   &    |-  G  =  ( n  e.  NN0  |->  if ( n  =  0 ,  (  .0.  .-  ( ( T `  M )  .X.  ( T `  ( b `
  0 ) ) ) ) ,  if ( n  =  (
 s  +  1 ) ,  ( T `  ( b `  s
 ) ) ,  if ( ( s  +  1 )  <  n ,  .0.  ,  ( ( T `
  ( b `  ( n  -  1
 ) ) )  .-  ( ( T `  M )  .X.  ( T `
  ( b `  n ) ) ) ) ) ) ) )   =>    |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B ) 
 /\  ( s  e. 
 NN  /\  b  e.  ( B  ^m  ( 0
 ... s ) ) ) )  ->  G : NN0 --> ( Base `  Y ) )
 
Theoremchfacfisfcpmat 19118* The "characteristic factor function" is a function from the nonnegative integers to constant polynomial matrices. (Contributed by AV, 19-Nov-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  P  =  (Poly1 `  R )   &    |-  Y  =  ( N Mat  P )   &    |-  .X. 
 =  ( .r `  Y )   &    |-  .-  =  ( -g `  Y )   &    |-  .0.  =  ( 0g `  Y )   &    |-  T  =  ( N matToPolyMat  R )   &    |-  G  =  ( n  e.  NN0  |->  if ( n  =  0 ,  (  .0.  .-  ( ( T `  M )  .X.  ( T `  ( b `
  0 ) ) ) ) ,  if ( n  =  (
 s  +  1 ) ,  ( T `  ( b `  s
 ) ) ,  if ( ( s  +  1 )  <  n ,  .0.  ,  ( ( T `
  ( b `  ( n  -  1
 ) ) )  .-  ( ( T `  M )  .X.  ( T `
  ( b `  n ) ) ) ) ) ) ) )   &    |-  S  =  ( N ConstPolyMat  R )   =>    |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  B ) 
 /\  ( s  e. 
 NN  /\  b  e.  ( B  ^m  ( 0
 ... s ) ) ) )  ->  G : NN0 --> S )
 
Theoremchfacffsupp 19119* The "characteristic factor function" is finitely supported. (Contributed by AV, 20-Nov-2019.) (Proof shortened by AV, 23-Dec-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  P  =  (Poly1 `  R )   &    |-  Y  =  ( N Mat  P )   &    |-  .X. 
 =  ( .r `  Y )   &    |-  .-  =  ( -g `  Y )   &    |-  .0.  =  ( 0g `  Y )   &    |-  T  =  ( N matToPolyMat  R )   &    |-  G  =  ( n  e.  NN0  |->  if ( n  =  0 ,  (  .0.  .-  ( ( T `  M )  .X.  ( T `  ( b `
  0 ) ) ) ) ,  if ( n  =  (
 s  +  1 ) ,  ( T `  ( b `  s
 ) ) ,  if ( ( s  +  1 )  <  n ,  .0.  ,  ( ( T `
  ( b `  ( n  -  1
 ) ) )  .-  ( ( T `  M )  .X.  ( T `
  ( b `  n ) ) ) ) ) ) ) )   =>    |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B ) 
 /\  ( s  e. 
 NN  /\  b  e.  ( B  ^m  ( 0
 ... s ) ) ) )  ->  G finSupp  ( 0g `  Y ) )
 
Theoremchfacfscmulcl 19120* Closure of a scaled value of the "characteristic factor function". (Contributed by AV, 9-Nov-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  P  =  (Poly1 `  R )   &    |-  Y  =  ( N Mat  P )   &    |-  .X. 
 =  ( .r `  Y )   &    |-  .-  =  ( -g `  Y )   &    |-  .0.  =  ( 0g `  Y )   &    |-  T  =  ( N matToPolyMat  R )   &    |-  G  =  ( n  e.  NN0  |->  if ( n  =  0 ,  (  .0.  .-  ( ( T `  M )  .X.  ( T `  ( b `
  0 ) ) ) ) ,  if ( n  =  (
 s  +  1 ) ,  ( T `  ( b `  s
 ) ) ,  if ( ( s  +  1 )  <  n ,  .0.  ,  ( ( T `
  ( b `  ( n  -  1
 ) ) )  .-  ( ( T `  M )  .X.  ( T `
  ( b `  n ) ) ) ) ) ) ) )   &    |-  X  =  (var1 `  R )   &    |-  .x.  =  ( .s `  Y )   &    |-  .^  =  (.g `  (mulGrp `  P )
 )   =>    |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B ) 
 /\  ( s  e. 
 NN  /\  b  e.  ( B  ^m  ( 0
 ... s ) ) )  /\  K  e.  NN0 )  ->  ( ( K  .^  X )  .x.  ( G `  K ) )  e.  ( Base `  Y ) )
 
Theoremchfacfscmul0 19121* A scaled value of the "characteristic factor function" is zero almost always. (Contributed by AV, 9-Nov-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  P  =  (Poly1 `  R )   &    |-  Y  =  ( N Mat  P )   &    |-  .X. 
 =  ( .r `  Y )   &    |-  .-  =  ( -g `  Y )   &    |-  .0.  =  ( 0g `  Y )   &    |-  T  =  ( N matToPolyMat  R )   &    |-  G  =  ( n  e.  NN0  |->  if ( n  =  0 ,  (  .0.  .-  ( ( T `  M )  .X.  ( T `  ( b `
  0 ) ) ) ) ,  if ( n  =  (
 s  +  1 ) ,  ( T `  ( b `  s
 ) ) ,  if ( ( s  +  1 )  <  n ,  .0.  ,  ( ( T `
  ( b `  ( n  -  1
 ) ) )  .-  ( ( T `  M )  .X.  ( T `
  ( b `  n ) ) ) ) ) ) ) )   &    |-  X  =  (var1 `  R )   &    |-  .x.  =  ( .s `  Y )   &    |-  .^  =  (.g `  (mulGrp `  P )
 )   =>    |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B ) 
 /\  ( s  e. 
 NN  /\  b  e.  ( B  ^m  ( 0
 ... s ) ) )  /\  K  e.  ( ZZ>= `  ( s  +  2 ) ) )  ->  ( ( K  .^  X )  .x.  ( G `  K ) )  =  .0.  )
 
Theoremchfacfscmulfsupp 19122* A mapping of scaled values of the "characteristic factor function" is finitely supported. (Contributed by AV, 8-Nov-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  P  =  (Poly1 `  R )   &    |-  Y  =  ( N Mat  P )   &    |-  .X. 
 =  ( .r `  Y )   &    |-  .-  =  ( -g `  Y )   &    |-  .0.  =  ( 0g `  Y )   &    |-  T  =  ( N matToPolyMat  R )   &    |-  G  =  ( n  e.  NN0  |->  if ( n  =  0 ,  (  .0.  .-  ( ( T `  M )  .X.  ( T `  ( b `
  0 ) ) ) ) ,  if ( n  =  (
 s  +  1 ) ,  ( T `  ( b `  s
 ) ) ,  if ( ( s  +  1 )  <  n ,  .0.  ,  ( ( T `
  ( b `  ( n  -  1
 ) ) )  .-  ( ( T `  M )  .X.  ( T `
  ( b `  n ) ) ) ) ) ) ) )   &    |-  X  =  (var1 `  R )   &    |-  .x.  =  ( .s `  Y )   &    |-  .^  =  (.g `  (mulGrp `  P )
 )   =>    |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B ) 
 /\  ( s  e. 
 NN  /\  b  e.  ( B  ^m  ( 0
 ... s ) ) ) )  ->  (
 i  e.  NN0  |->  ( ( i  .^  X )  .x.  ( G `  i
 ) ) ) finSupp  .0.  )
 
Theoremchfacfscmulgsum 19123* Breaking up a sum of values of the "characteristic factor function" scaled by a polynomial. (Contributed by AV, 9-Nov-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  P  =  (Poly1 `  R )   &    |-  Y  =  ( N Mat  P )   &    |-  .X. 
 =  ( .r `  Y )   &    |-  .-  =  ( -g `  Y )   &    |-  .0.  =  ( 0g `  Y )   &    |-  T  =  ( N matToPolyMat  R )   &    |-  G  =  ( n  e.  NN0  |->  if ( n  =  0 ,  (  .0.  .-  ( ( T `  M )  .X.  ( T `  ( b `
  0 ) ) ) ) ,  if ( n  =  (
 s  +  1 ) ,  ( T `  ( b `  s
 ) ) ,  if ( ( s  +  1 )  <  n ,  .0.  ,  ( ( T `
  ( b `  ( n  -  1
 ) ) )  .-  ( ( T `  M )  .X.  ( T `
  ( b `  n ) ) ) ) ) ) ) )   &    |-  X  =  (var1 `  R )   &    |-  .x.  =  ( .s `  Y )   &    |-  .^  =  (.g `  (mulGrp `  P )
 )   &    |- 
 .+  =  ( +g  `  Y )   =>    |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B ) 
 /\  ( s  e. 
 NN  /\  b  e.  ( B  ^m  ( 0
 ... s ) ) ) )  ->  ( Y  gsumg  ( i  e.  NN0  |->  ( ( i  .^  X )  .x.  ( G `
  i ) ) ) )  =  ( ( Y  gsumg  ( i  e.  (
 1 ... s )  |->  ( ( i  .^  X )  .x.  ( ( T `
  ( b `  ( i  -  1
 ) ) )  .-  ( ( T `  M )  .X.  ( T `
  ( b `  i ) ) ) ) ) ) ) 
 .+  ( ( ( ( s  +  1 )  .^  X )  .x.  ( T `  (
 b `  s )
 ) )  .-  (
 ( T `  M )  .X.  ( T `  ( b `  0
 ) ) ) ) ) )
 
Theoremchfacfpmmulcl 19124* Closure of the value of the "characteristic factor function" multiplied with a constant polynomial matrix. (Contributed by AV, 23-Nov-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  P  =  (Poly1 `  R )   &    |-  Y  =  ( N Mat  P )   &    |-  .X. 
 =  ( .r `  Y )   &    |-  .-  =  ( -g `  Y )   &    |-  .0.  =  ( 0g `  Y )   &    |-  T  =  ( N matToPolyMat  R )   &    |-  G  =  ( n  e.  NN0  |->  if ( n  =  0 ,  (  .0.  .-  ( ( T `  M )  .X.  ( T `  ( b `
  0 ) ) ) ) ,  if ( n  =  (
 s  +  1 ) ,  ( T `  ( b `  s
 ) ) ,  if ( ( s  +  1 )  <  n ,  .0.  ,  ( ( T `
  ( b `  ( n  -  1
 ) ) )  .-  ( ( T `  M )  .X.  ( T `
  ( b `  n ) ) ) ) ) ) ) )   &    |-  .^  =  (.g `  (mulGrp `  Y ) )   =>    |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B ) 
 /\  ( s  e. 
 NN  /\  b  e.  ( B  ^m  ( 0
 ... s ) ) )  /\  K  e.  NN0 )  ->  ( ( K  .^  ( T `  M ) )  .X.  ( G `  K ) )  e.  ( Base `  Y ) )
 
Theoremchfacfpmmul0 19125* The value of the "characteristic factor function" multiplied with a constant polynomial matrix is zero almost always. (Contributed by AV, 23-Nov-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  P  =  (Poly1 `  R )   &    |-  Y  =  ( N Mat  P )   &    |-  .X. 
 =  ( .r `  Y )   &    |-  .-  =  ( -g `  Y )   &    |-  .0.  =  ( 0g `  Y )   &    |-  T  =  ( N matToPolyMat  R )   &    |-  G  =  ( n  e.  NN0  |->  if ( n  =  0 ,  (  .0.  .-  ( ( T `  M )  .X.  ( T `  ( b `
  0 ) ) ) ) ,  if ( n  =  (
 s  +  1 ) ,  ( T `  ( b `  s
 ) ) ,  if ( ( s  +  1 )  <  n ,  .0.  ,  ( ( T `
  ( b `  ( n  -  1
 ) ) )  .-  ( ( T `  M )  .X.  ( T `
  ( b `  n ) ) ) ) ) ) ) )   &    |-  .^  =  (.g `  (mulGrp `  Y ) )   =>    |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B ) 
 /\  ( s  e. 
 NN  /\  b  e.  ( B  ^m  ( 0
 ... s ) ) )  /\  K  e.  ( ZZ>= `  ( s  +  2 ) ) )  ->  ( ( K  .^  ( T `  M ) )  .X.  ( G `  K ) )  =  .0.  )
 
Theoremchfacfpmmulfsupp 19126* A mapping of values of the "characteristic factor function" multiplied with a constant polynomial matrix is finitely supported. (Contributed by AV, 23-Nov-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  P  =  (Poly1 `  R )   &    |-  Y  =  ( N Mat  P )   &    |-  .X. 
 =  ( .r `  Y )   &    |-  .-  =  ( -g `  Y )   &    |-  .0.  =  ( 0g `  Y )   &    |-  T  =  ( N matToPolyMat  R )   &    |-  G  =  ( n  e.  NN0  |->  if ( n  =  0 ,  (  .0.  .-  ( ( T `  M )  .X.  ( T `  ( b `
  0 ) ) ) ) ,  if ( n  =  (
 s  +  1 ) ,  ( T `  ( b `  s
 ) ) ,  if ( ( s  +  1 )  <  n ,  .0.  ,  ( ( T `
  ( b `  ( n  -  1
 ) ) )  .-  ( ( T `  M )  .X.  ( T `
  ( b `  n ) ) ) ) ) ) ) )   &    |-  .^  =  (.g `  (mulGrp `  Y ) )   =>    |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B ) 
 /\  ( s  e. 
 NN  /\  b  e.  ( B  ^m  ( 0
 ... s ) ) ) )  ->  (
 i  e.  NN0  |->  ( ( i  .^  ( T `  M ) )  .X.  ( G `  i ) ) ) finSupp  .0.  )
 
Theoremchfacfpmmulgsum 19127* Breaking up a sum of values of the "characteristic factor function" multiplied with a constant polynomial matrix. (Contributed by AV, 23-Nov-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  P  =  (Poly1 `  R )   &    |-  Y  =  ( N Mat  P )   &    |-  .X. 
 =  ( .r `  Y )   &    |-  .-  =  ( -g `  Y )   &    |-  .0.  =  ( 0g `  Y )   &    |-  T  =  ( N matToPolyMat  R )   &    |-  G  =  ( n  e.  NN0  |->  if ( n  =  0 ,  (  .0.  .-  ( ( T `  M )  .X.  ( T `  ( b `
  0 ) ) ) ) ,  if ( n  =  (
 s  +  1 ) ,  ( T `  ( b `  s
 ) ) ,  if ( ( s  +  1 )  <  n ,  .0.  ,  ( ( T `
  ( b `  ( n  -  1
 ) ) )  .-  ( ( T `  M )  .X.  ( T `
  ( b `  n ) ) ) ) ) ) ) )   &    |-  .^  =  (.g `  (mulGrp `  Y ) )   &    |-  .+  =  ( +g  `  Y )   =>    |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B ) 
 /\  ( s  e. 
 NN  /\  b  e.  ( B  ^m  ( 0
 ... s ) ) ) )  ->  ( Y  gsumg  ( i  e.  NN0  |->  ( ( i  .^  ( T `  M ) )  .X.  ( G `  i ) ) ) )  =  ( ( Y  gsumg  ( i  e.  (
 1 ... s )  |->  ( ( i  .^  ( T `  M ) ) 
 .X.  ( ( T `
  ( b `  ( i  -  1
 ) ) )  .-  ( ( T `  M )  .X.  ( T `
  ( b `  i ) ) ) ) ) ) ) 
 .+  ( ( ( ( s  +  1 )  .^  ( T `  M ) )  .X.  ( T `  ( b `
  s ) ) )  .-  ( ( T `  M )  .X.  ( T `  ( b `
  0 ) ) ) ) ) )
 
Theoremchfacfpmmulgsum2 19128* Breaking up a sum of values of the "characteristic factor function" multiplied with a constant polynomial matrix. (Contributed by AV, 23-Nov-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  P  =  (Poly1 `  R )   &    |-  Y  =  ( N Mat  P )   &    |-  .X. 
 =  ( .r `  Y )   &    |-  .-  =  ( -g `  Y )   &    |-  .0.  =  ( 0g `  Y )   &    |-  T  =  ( N matToPolyMat  R )   &    |-  G  =  ( n  e.  NN0  |->  if ( n  =  0 ,  (  .0.  .-  ( ( T `  M )  .X.  ( T `  ( b `
  0 ) ) ) ) ,  if ( n  =  (
 s  +  1 ) ,  ( T `  ( b `  s
 ) ) ,  if ( ( s  +  1 )  <  n ,  .0.  ,  ( ( T `
  ( b `  ( n  -  1
 ) ) )  .-  ( ( T `  M )  .X.  ( T `
  ( b `  n ) ) ) ) ) ) ) )   &    |-  .^  =  (.g `  (mulGrp `  Y ) )   &    |-  .+  =  ( +g  `  Y )   =>    |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B ) 
 /\  ( s  e. 
 NN  /\  b  e.  ( B  ^m  ( 0
 ... s ) ) ) )  ->  ( Y  gsumg  ( i  e.  NN0  |->  ( ( i  .^  ( T `  M ) )  .X.  ( G `  i ) ) ) )  =  ( ( Y  gsumg  ( i  e.  (
 1 ... s )  |->  ( ( ( i  .^  ( T `  M ) )  .X.  ( T `  ( b `  (
 i  -  1 ) ) ) )  .-  ( ( ( i  +  1 )  .^  ( T `  M ) )  .X.  ( T `  ( b `  i
 ) ) ) ) ) )  .+  (
 ( ( ( s  +  1 )  .^  ( T `  M ) )  .X.  ( T `  ( b `  s
 ) ) )  .-  ( ( T `  M )  .X.  ( T `
  ( b `  0 ) ) ) ) ) )
 
Theoremcayhamlem1 19129* Lemma 1 for cayleyhamilton 19153. (Contributed by AV, 11-Nov-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  P  =  (Poly1 `  R )   &    |-  Y  =  ( N Mat  P )   &    |-  .X. 
 =  ( .r `  Y )   &    |-  .-  =  ( -g `  Y )   &    |-  .0.  =  ( 0g `  Y )   &    |-  T  =  ( N matToPolyMat  R )   &    |-  G  =  ( n  e.  NN0  |->  if ( n  =  0 ,  (  .0.  .-  ( ( T `  M )  .X.  ( T `  ( b `
  0 ) ) ) ) ,  if ( n  =  (
 s  +  1 ) ,  ( T `  ( b `  s
 ) ) ,  if ( ( s  +  1 )  <  n ,  .0.  ,  ( ( T `
  ( b `  ( n  -  1
 ) ) )  .-  ( ( T `  M )  .X.  ( T `
  ( b `  n ) ) ) ) ) ) ) )   &    |-  .^  =  (.g `  (mulGrp `  Y ) )   =>    |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B ) 
 /\  ( s  e. 
 NN  /\  b  e.  ( B  ^m  ( 0
 ... s ) ) ) )  ->  ( Y  gsumg  ( i  e.  NN0  |->  ( ( i  .^  ( T `  M ) )  .X.  ( G `  i ) ) ) )  =  .0.  )
 
11.5.3  The Cayley-Hamilton theorem

In this section, a direct algebraic proof for the Cayley-Hamilton theorem is provided, according to Wikipedia ("Cayley-Hamilton theorem", 09-Nov-2019, https://en.wikipedia.org/wiki/Cayley%E2%80%93Hamilton_theorem, section "A direct algebraic proof" (this approach is also used for proving Lemma 1.9 in [Hefferon] p. 427):

"This proof uses just the kind of objects needed to formulate the Cayley-Hamilton theorem: matrices with polynomials as entries. The matrix (t * In - A) whose determinant is the characteristic polynomial of A is such a matrix, and since polynomials [over a commutative ring] form a commutative ring, it has an adjugate

(1) B = adj(t * In - A) .

Then, according to the right-hand fundamental relation of the adjugate, one has

(2) ( t * In - A ) x B = det(t * In - A) x In = p(t) * In .

Since B is also a matrix with polynomials in t as entries, one can, for each i, collect the coefficients of t^i in each entry to form a matrix Bi of numbers, such that one has

(3) B = sumi = 0 to (n-1) t^i Bi .

(The way the entries of B are defined makes clear that no powers higher than t^(n-1) occur). While this looks like a polynomial with matrices as coefficients, we shall not consider such a notion; it is just a way to write a matrix with polynomial entries as a linear combination of n constant matrices, and the coefficient t^i has been written to the left of the matrix to stress this point of view.

Now, one can expand the matrix product in our equation by bilinearity

(4) p(t) * In = ( t * In - A ) x B
= ( t * In - A ) x sumi = 0 to (n-1) t^i * Bi
= sumi = 0 to (n-1) t * In x t^i Bi - sumi = 0 to (n-1) A * t^i Bi
= sumi = 0 to (n-1) t^(i+1) * Bi - sumi = 0 to (n-1) t^i * A x Bi
= t^n Bn-1 + sumi = 1 to (n-1) t^i * ( Bi-1 - A x Bi ) - A x B0 .

Writing

(5) p(t) In = t^n * In + t^(n-1) * c(n-1) x In + ... + t * c1 In + c0 In ,

one obtains an equality of two matrices with polynomial entries, written as linear combinations of constant matrices with powers of t as coefficients. Such an equality can hold only if in any matrix position the entry that is multiplied by a given power t^i is the same on both sides; it follows that the constant matrices with coefficient t^i in both expressions must be equal. Writing these equations then for i from n down to 0, one finds

(6) Bn-1 = In , Bi-1 - A x Bi = ci * In for 1 <= i <= n-1 , - A x B0 = c0 * In .

Finally, multiply the equation of the coefficients of t^i from the left by A^i, and sum up:

(7) A^n Bn-1 + sumi = 1 to (n-1) ( A^i x Bi-1 - A^(i+1) x Bi ) - A x B0 = A^n + cn-1 * A^(n-1) + ... + c1 * A + c0 * In .

The left-hand sides form a telescoping sum and cancel completely; the right-hand sides add up to p(A):

(8) 0 = p(A) .

This completes the proof."

To formalize this approach, the steps mentioned in Wikipedia must be elaborated in more detail.

The first step is to formalize the preliminaries and the objective of the theorem. In Wikipedia, the Cayley-Hamilton theorem is stated as follows: "... the Cayley-Hamilton theorem ... states that every square matrix over a commutative ring ... satisfies its own characteristic equation." Or in more detail: "If A is a given n x n matrix and In is the n x n identity matrix, then the characteristic polynomial of A is defined as p(t) = det(t * In - A), where det is the determinant operation and t is a variable for a scalar element of the base ring. Since the entries of the matrix (t * In - A) are (linear or constant) polynomials in t, the determinant is also an n-th order monic polynomial in t. The Cayley-Hamilton theorem states that if one defines an analogous matrix equation, p(A), consisting of the replacement of the scalar eigenvalues t with the matrix A, then this polynomial in the matrix A results in the zero matrix,

p(A) = 0.

The powers of A, obtained by substitution from powers of t, are defined by repeated matrix multiplication; the constant term of p(t) gives a multiple of the power A^0, which is defined as the identity matrix. The theorem allows A^n to be expressed as a linear combination of the lower matrix powers of A. When the ring is a field, the Cayley-Hamilton theorem is equivalent to the statement that the minimal polynomial of a square matrix divides its characteristic polynomial."

Actually, the definition of the characteristic polynomial of a square matrix requires some attention. According to df-chpmat 19090, the characteristic polynomial of an  N x  N matrix  M over a ring  R is defined as

 ( ( N CharPlyMat  R ) `  M )  =  ( D `  ( ( X  .x.  .1.  )  .-  ( T `  M ) ) ) )

where  D  =  ( N maDet  P ) is the function mapping an  N x  N matrix over the polynomial ring over the ring  R to its determinant,  X  =  (var1 `  R ) is the variable of the polynomials over  R,  .1. is the  N x  N identity matrix as matrix over the polynomial ring over the ring  R (not the  N x  N identity matrix of the matrices over the ring  R!) and  ( T `  M )  =  ( ( N matToPolyMat  R ) `  M ) is the matrix  M over a ring  R transformed into a constant matrix over the polynomial ring over the ring  R. Thus  .x. is the multiplication of a polynomial matrix with a "scalar", i.e. a polynomial (see chpmatval 19094).

By this definition, it is assured that  ( ( X  .x.  .1.  )  .-  ( T `  M ) ), the matrix whose determinat is the characteristic polynomial of the matrix  M, is actually a matrix over the polynomial ring over the ring  R, as stated in Wikipedia ("matrix with polynomials as entries"). This matrix is called the characteristic matrix of matrix  M (see Wikipedia "Polynomial matrix", 16-Nov-2019, https://en.wikipedia.org/wiki/Polynomial_matrix). Following the notation in Wikipedia, we denote the characteristic polynomial of the matrix  M, formally defined by  ( ( N CharPlyMat  R ) `  M ) as "p(M)" in the comments. The characteristric matrix  ( ( X  .x.  .1.  )  .-  ( T `  M ) ) will be denoted by "c(M)", so that "p(M) = det( c(M) )".

After the preliminaries are clarified, the objective of the Cayley-Hamilton theorem must be considered. As described in Wikipedia, the matrix  M must be "inserted" into its characteristic polynomial in an appropriate way. Since every polynomial can be represented as infinite, but finitely supported sum of monomials scaled by the corresponding coefficients (see ply1coe 18103), also the characteristic polynomial can be written in this way:

p(M) = sumi ( pi * M^i )

Here, * is the scalar multiplication in the algebra of the polynomials over the ring  R, and the coefficients are elements of the ring  R.

By this, we can "define" the insertion of the matrix M into its characteristic polynomial by "p(M) = sum( pi * M^i)", see also cayleyhamilton1 19155. Here, * is the scalar multiplication in the algebra of the matrices over the ring  R.

To prove the Cayley-Hamilton theorem, we have to show that "p(M) = 0", where 0 is the zero matrix.

In this section, the following class variables and informal identifiers (acronyms in the form "A(B)" or "AB") are used:

class variable/ informal identifier definiens explanation
 N An arbitrary finite set, used as dimension for matrices
 R An arbitrary (commutative) ring:  R  e.  CRing
B(R)  ( Base `  R ) Base set of (the ring)  R
 A  ( N Mat  R ) Algebra of  N x  N matrices over (the ring)  R
 B  ( Base `  A ) Base set of the algebra of  N x  N matrices, i .e. the set of all  N x  N matrices
 M An arbitrary  N x  N matrix
 P  (Poly1 `  R ) The algebra of polynomials over (the ring)  R
B(P)  ( Base `  P ) Base set of the algebra of polynomials, i .e. the set of all polynomials
 X, XR  (var1 `  R ) The variable of polynomials over (the ring)  R
 Y, XA  (var1 `  A ) The variable of polynomials over matrices of the algebra  A
 .^  (.g `  (mulGrp `  P ) ) The group exponentiation for polynomials over (the ring)  R
^ Arbitrary group exponentiation
 U  (algSc `  P ) The injection of scalars, i.e. elements of (the ring)  R into the base set of the algebra of polynomials over  R
 ( U `  p ), S(p)  ( (algSc `  P ) `  p ) The element  p of (the ring)  R represented as polynomial over  R
 Y  ( N Mat  P ) Algebra of  N x  N matrices over (the polynomial ring)  P over the ring  R
B(Y)  ( Base `  Y ) Base set of the algebra of polynomial  N x  N matrices, i .e. the set of all polynomial  N x  N matrices
 Q  (Poly1 `  A ) Algebra of polynomials over the ring of  N x  N matrices over the ring  R
B(Q)  ( Base `  Q ) Base set of the algebra of polynomials over the ring of  N x  N matrices over the ring  R, i .e. the set of all polynomials having  N x  N matrices as coefficients
 .+, +  ( +g  `  Y ) The addition of polynomial matrices
 .-, -  ( -g `  Y ) The subtraction of polynomial matrices
 .x., *Y  ( .s `  Y ) The multiplication of a polynomial matrix with a scalar ( i. e. a polynomial)
*A  ( .s `  A ) The multiplication of a matrix with a scalar ( i. e. an element of the underlying ring)
*Q  ( .s `  Q ) The multiplication of a polynomial over matrices with a scalar ( i. e. a matrix)
 .X., xY  ( .r `  Y ) The multiplication of polynomial matrices
xA  ( .r `  A ) The multiplication of matrices
xQ  ( .r `  Q ) The multiplication of polynomials over matrices
 .1., 1Y  ( 1r `  Y ) The identity matrix in the algebra of polynomial matrices over  R
1A  ( 1r `  A ) The identity matrix in the algebra of matrices over  R
 .0., 0Y  ( 0g `  Y ) The zero matrix in the algebra of matrices consisting of polynomials
 T  ( N matToPolyMat  R ) The transformation of an  N x  N matrix over  R into a polynomial  N x  N matrix over  R
T1(M)  ( T `  M ) The matrix M transformed into a polynomial  N x  N matrix over  R
U(M)  ( U `  M ) The (constant) polynomial  N x  N matrix M transformed into a matrix over the ring  R. Inverse function of  T:  ( T `  ( U `  M ) )  =  M (see m2cpminvid2 19018 )
T2(M)  ( ( N pMatToMatPoly  R ) `  M ) The polynomial  N x  N matrix M transformed into a polynomial over the  N x  N matrices over  R
 I, c(M)  ( ( X  .x.  .1.  )  .-  ( T `  M ) ) The characteristic matrix of a matrix  M, i.e. the matrix whose determinant is the characteristic polynomial of the matrix  M
 C  ( N CharPlyMat  R ) The function mapping a matrix over (a ring)  R to its characteristic polynomial
 K, p(M)  ( C `  M ) The characteristic polynomial of a matrix over (a ring)  R
 H  ( K  .x.  .1.  ) The scalar matrix (diagonal matrix) with the characteristic polynomial of a matrix as diagional elements
 J  ( N maAdju  P ) The function mapping a matrix consisting of polynomials to its adjugate ("matrix of cofactors")
 W, adj(cm(M))  ( J `  I ) The adjugate of the characteristic matrix of the matrix  M


Using this notation, we have:
  1. "c(M) e. B(Y)", or  I  e.  ( Base `  Y ), see chmatcl 19091
  2. "p(M) e. B(P)", or  K  e.  ( Base `  P ), see chpmatply1 19095
  3. "T(M) e. B(Y)", or  ( T `  M )  e.  ( Base `  Y ), see mat2pmatbas 18989
  4.  J : ( Base `  Y ) --> ( Base `  Y ), see maduf 18905
  5. "adj(cm(M)) e. B(Y)", or  W  e.  ( Base `  Y )


Following the proof shown in Wikipedia, the following steps are performed:
  1. Write  W, the adjugate of the characteristic matrix, as a finite sum of scaled monomials, see pmatcollpw3fi1 19051:
    adj(cm(M)) = sumi=0 to s ( XR ^i *Y T1(b(i)) )
    where b(i) are matrices over the ring  R, so T1(b(i)) are constant polynomial matrices.
    This step corresponds to (3) in Wikipedia. In contrast to Wikipedia, we write  W as finite sum of not exactly determined number of summands, which may be greater than needed (including summands of value 0). This will be sufficient to provide a representation of  ( I  .X.  W ) as infinite, but finitely supported sum, see step 3.
  2. Write  ( I  .X.  W ), the product of the characteristic matrix and its adjugate as finite sum of scaled monomials, see cpmadugsumfi 19140. This representation is obtained by replacing  W by the representation resulting from step 1. and performing calculation rules available for the associative algebra of matrices over polynomials over a commutative ring:
    cm(M) *Y adj(cm(M)) = sumi=0 to s ( XR ^i *Y ( T1(b(i-1)) - T1(M) xY T1(b(i)) ) ) + XR ^(s+1) *Y ( T1(b(s)) - T1(M) xY T1(b(0))
    where b(i) are matrices over  R, so T1(b(i)) are constant polynomial matrices:
    cm(M) *Y adj(cm(M))
    = cm(M) *Y sumi=0 to s ( XR ^i *Y T1(b(i)) ) [see pmatcollpw3fi1 19051 (step 1.)]
    = ( ( XA *Y 1Y ) - T1(M) ) *Y sumi=0 to s ( XR ^i *Y T1(b(i)) ) [def. of cm(M)]
    = ( XA *Y 1Y ) *Y sumi=0 to s ( XR ^i *Y T1(b(i)) ) - T1(M) *Y sumi=0 to s ( XR ^i *Y T1(b(i)) ) [see rngsubdir 17027]
    = sumi=0 to s ( XR ^i *Y ( T1(b(i-1)) - T1(M) xY T1(b(i)) ) ) + XR ^(s+1) *Y ( T1(b(s)) - T1(M) xY T1(b(0)) [see cpmadugsumlemF 19139]
    This step corresponds partially to (4) in Wikipedia.
  3. Write  ( I  .X.  W ) as infinite, but finitely supported sum of scaled monomials, see cpmadugsum 19141:
    cm(M) * adj(cm(M)) = sumi ( XR ^i *Y G(i) )
    This representation is obtained by defining a function G for the coefficients, which we call "characteristic factor function", see chfacfisf 19117, which covers the special terms and the padding with 0. G(i) is a constant polynomial matrix (see chfacfisfcpmat 19118). This step corresponds partially to (4) in Wikipedia, with summands of value 0 added.
  4. Write  H  =  ( K  .x.  .1.  ), the scalar matrix (diagonal matrix) with the characteristic polynomial of a matrix as diagional elements, as infinite, but finitely supported sum of scaled monomials. See cpmidgsum 19131:
    p(m) *Y IY = sumi ( XR ^i *Y ( S(pi) *Y IY ) )
    The proof of cpmidgsum 19131 is making use of pmatcollpwscmat 19054, because  H  =  ( K  .x.  .1.  ) is a scalar/diagonal polynomial matrix with the characteristic polynomial "p(M)" as diagonal entries (since pi is an element of the ring  R, S(pi) is a (constant) polynomial). This corresponds to (5) in Wikipedia, with summands of value 0 added.
  5. Transform the sum representation of  ( I  .X.  W ) from step 3. into polynomials over matrices:
    T2(cm(M) * adj(cm(M))) = sumi ( U(G(i)) *Q XA ^i ) [see cpmadumatpoly 19146]
    where U(G(i)) is a matrix over the ring  R.
  6. Transform the sum representation of  H from step 4. into polynomials over matrices:
    T2(p(m) *Y IY) = sumi ( pi *A IA ) *Q XA ^i ) [see cpmidpmat 19136]
  7. Equate the sum representations resulting from steps 5. and 6. by using cpmadurid 19130 to obtain the equation
    sumi ( U(G(i)) *Q XA ^i ) = sumi ( pi *A IA ) *Q XA ^i ):
    sumi ( U(G(i)) *Q XA ^i )
    = T2(cm(M) * adj(cm(M))) [see step 5.]
    = T2(p(m) *Y IY) [see cpmadurid 19130]
    = sumi ( pi *A IA ) *Q XA ^i ) [see step 6.]
    Note that this step is contained in the proof of chcoeffeq 19149, see step 9. This step corresponds to the conclusion from (4) and (5) in Wikipedia, with summands of value 0 added.
  8. Compare the sum representations of step 7. to obtain the equations U(G(i)) = pi *A IA , see chcoeffeqlem 19148. This corresponds to (6) in Wikipedia. Since the coefficients of the transformed representations and the original representations are identical, the equations of the coefficients are also valid for the original representations of steps 3. and 4.
  9. Multiply the equations of the coefficients from step 8. from the left by M^i, and sum up, see chcoeffeq 19149:
    sumi ( M^i xA U(G(i)) ) = sumi ( M^i xA ( pi *A IA) )
    This corresponds to (7) in Wikipedia.
  10. Transform the right hand side of the equation in step 9. into an appropriate form, see cayhamlem3 19150:
    sumi ( pi *A M^i )
    = sumi ( M^i xA ( pi *A IA) ) [see cayhamlem2 19147]
    = sumi ( M^i xA U(G(i)) ) [see chcoeffeq 19149]
  11. Apply the theorem for telescoping sums, see telgsumfz 16805, to the sum sumi ( T1(M)^i xY G(i) ), which results in an equation to 0:
    sumi ( T1(M)^i xY G(i) ) = 0Y, see cayhamlem1 19129:
    sumi ( T1(M)^i xY G(i) )
    = sumi=1 to s ( T1(M)^i xY T1(b(i-1)) - T1(M)^(i+1) xY T1(b(i)) )
    + ( T1(M)^(s+1) xY T1(b(s)) - T1(M) xY T1(b(0)) ) [see chfacfpmmulgsum2 19128]
    = ( T1(M) xY T1(b(0)) - T1(M)^(s+1) xY T1(b(s)) ) + ( T1 M)^(s+1) xY T1(b(s)) - T1(M) xY T1(b(0)) ) [see telgsumfz 16805]
    = 0Y [see grpnpncan0 15930] This step corresponds partially to (8) in Wikipedia.
  12. Since  T is a ring homomorphism (see mat2pmatrhm 18997), the left hand side of the equation in step 10. can be transformed into a representation appropriate to apply the result of step 11., see cayhamlem4 19151:
    sumi ( pi *A M^i )
    = sumi ( M^i xA U(G(i)) ) [see cayhamlem3 19150 (step 10.)]
    = U(T1(sumi ( M^i xA U(G(i)) ))) [see m2cpminvid 19016]
    = U(sumi T1( M^i xA U(G(i)) )) [see gsummptmhm 16749]
    = U(sumi ( T1(M^i) xY T1(U(G(i))) )) [see rhmmul 17155]
    = U(sumi ( T1(M)^i xY T1(U(G(i))) )) [see mhmmulg 15969]
    = U(sumi ( T1(M)^i xY G(i) )) [see m2cpminvid2 19018 ]
  13. Finally, combine the results of steps 11. and 12., and use the fact that  T (and therefore also its inverse  U) is an injective ring homomorphism (see mat2pmatf1 18992 and mat2pmatrhm 18997) to transform the equality resulting from steps 11. and 12. into the desired equation sumi ( pi *A M^i ) = 0A , see cayleyhamilton 19153 resp. cayleyhamilton0 19152:
    sumi ( pi *A M^i )
    = U(sumi ( T1(M)^i xY G(i) )) [see cayhamlem4 19151 (step 12.)]
    = U(0Y ) [see cayhamlem1 19129 (step 11.)]
    = 0A [see m2cpminv0 19024]
The transformations in steps 5., 6., 10., 12. and 13. are not mentioned in the proof provided in Wikipedia, since it makes no distinction between a matrix over a ring  R and its representation as matrix over the polynomial ring over the ring  R in general!
 
Theoremcpmadurid 19130 The right-hand fundamental relation of the adjugate (see madurid 18908) applied to the characteristic matrix of a matrix. (Contributed by AV, 25-Oct-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  C  =  ( N CharPlyMat  R )   &    |-  P  =  (Poly1 `  R )   &    |-  Y  =  ( N Mat  P )   &    |-  X  =  (var1 `  R )   &    |-  T  =  ( N matToPolyMat  R )   &    |-  .-  =  ( -g `  Y )   &    |-  .x.  =  ( .s `  Y )   &    |-  .1.  =  ( 1r `  Y )   &    |-  I  =  ( ( X  .x.  .1.  )  .-  ( T `  M ) )   &    |-  J  =  ( N maAdju  P )   &    |-  .X.  =  ( .r `  Y )   =>    |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  ( I  .X.  ( J `  I ) )  =  ( ( C `  M )  .x.  .1.  )
 )
 
Theoremcpmidgsum 19131* Representation of the identity matrix multiplied with the characteristic polynomial of a matrix as group sum. (Contributed by AV, 7-Nov-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  P  =  (Poly1 `  R )   &    |-  Y  =  ( N Mat  P )   &    |-  X  =  (var1 `  R )   &    |-  .^  =  (.g `  (mulGrp `  P ) )   &    |-  .x.  =  ( .s `  Y )   &    |-  .1.  =  ( 1r `  Y )   &    |-  U  =  (algSc `  P )   &    |-  C  =  ( N CharPlyMat  R )   &    |-  K  =  ( C `  M )   &    |-  H  =  ( K  .x.  .1.  )   =>    |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  H  =  ( Y  gsumg  ( n  e.  NN0  |->  ( ( n  .^  X )  .x.  ( ( U `  ( (coe1 `  K ) `  n ) )  .x.  .1.  )
 ) ) ) )
 
Theoremcpmidgsumm2pm 19132* Representation of the identity matrix multiplied with the characteristic polynomial of a matrix as group sum with a matrix to polynomial matrix transformation. (Contributed by AV, 13-Nov-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  P  =  (Poly1 `  R )   &    |-  Y  =  ( N Mat  P )   &    |-  X  =  (var1 `  R )   &    |-  .^  =  (.g `  (mulGrp `  P ) )   &    |-  .x.  =  ( .s `  Y )   &    |-  .1.  =  ( 1r `  Y )   &    |-  U  =  (algSc `  P )   &    |-  C  =  ( N CharPlyMat  R )   &    |-  K  =  ( C `  M )   &    |-  H  =  ( K  .x.  .1.  )   &    |-  O  =  ( 1r `  A )   &    |-  .*  =  ( .s `  A )   &    |-  T  =  ( N matToPolyMat  R )   =>    |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  H  =  ( Y  gsumg  ( n  e.  NN0  |->  ( ( n  .^  X )  .x.  ( T `
  ( ( (coe1 `  K ) `  n )  .*  O ) ) ) ) ) )
 
Theoremcpmidpmatlem1 19133* Lemma 1 for cpmidpmat 19136. (Contributed by AV, 13-Nov-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  P  =  (Poly1 `  R )   &    |-  Y  =  ( N Mat  P )   &    |-  X  =  (var1 `  R )   &    |-  .^  =  (.g `  (mulGrp `  P ) )   &    |-  .x.  =  ( .s `  Y )   &    |-  .1.  =  ( 1r `  Y )   &    |-  U  =  (algSc `  P )   &    |-  C  =  ( N CharPlyMat  R )   &    |-  K  =  ( C `  M )   &    |-  H  =  ( K  .x.  .1.  )   &    |-  O  =  ( 1r `  A )   &    |-  .*  =  ( .s `  A )   &    |-  T  =  ( N matToPolyMat  R )   &    |-  G  =  ( k  e.  NN0  |->  ( ( (coe1 `  K ) `  k )  .*  O ) )   =>    |-  ( L  e.  NN0  ->  ( G `  L )  =  ( ( (coe1 `  K ) `  L )  .*  O ) )
 
Theoremcpmidpmatlem2 19134* Lemma 2 for cpmidpmat 19136. (Contributed by AV, 14-Nov-2019.) (Proof shortened by AV, 7-Dec-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  P  =  (Poly1 `  R )   &    |-  Y  =  ( N Mat  P )   &    |-  X  =  (var1 `  R )   &    |-  .^  =  (.g `  (mulGrp `  P ) )   &    |-  .x.  =  ( .s `  Y )   &    |-  .1.  =  ( 1r `  Y )   &    |-  U  =  (algSc `  P )   &    |-  C  =  ( N CharPlyMat  R )   &    |-  K  =  ( C `  M )   &    |-  H  =  ( K  .x.  .1.  )   &    |-  O  =  ( 1r `  A )   &    |-  .*  =  ( .s `  A )   &    |-  T  =  ( N matToPolyMat  R )   &    |-  G  =  ( k  e.  NN0  |->  ( ( (coe1 `  K ) `  k )  .*  O ) )   =>    |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  G  e.  ( B  ^m  NN0 ) )
 
Theoremcpmidpmatlem3 19135* Lemma 3 for cpmidpmat 19136. (Contributed by AV, 14-Nov-2019.) (Proof shortened by AV, 7-Dec-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  P  =  (Poly1 `  R )   &    |-  Y  =  ( N Mat  P )   &    |-  X  =  (var1 `  R )   &    |-  .^  =  (.g `  (mulGrp `  P ) )   &    |-  .x.  =  ( .s `  Y )   &    |-  .1.  =  ( 1r `  Y )   &    |-  U  =  (algSc `  P )   &    |-  C  =  ( N CharPlyMat  R )   &    |-  K  =  ( C `  M )   &    |-  H  =  ( K  .x.  .1.  )   &    |-  O  =  ( 1r `  A )   &    |-  .*  =  ( .s `  A )   &    |-  T  =  ( N matToPolyMat  R )   &    |-  G  =  ( k  e.  NN0  |->  ( ( (coe1 `  K ) `  k )  .*  O ) )   =>    |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  G finSupp  ( 0g `  A ) )
 
Theoremcpmidpmat 19136* Representation of the identity matrix multiplied with the characteristic polynomial of a matrix as polynomial over the ring of matrices. (Contributed by AV, 14-Nov-2019.) (Revised by AV, 7-Dec-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  P  =  (Poly1 `  R )   &    |-  Y  =  ( N Mat  P )   &    |-  X  =  (var1 `  R )   &    |-  .^  =  (.g `  (mulGrp `  P ) )   &    |-  .x.  =  ( .s `  Y )   &    |-  .1.  =  ( 1r `  Y )   &    |-  U  =  (algSc `  P )   &    |-  C  =  ( N CharPlyMat  R )   &    |-  K  =  ( C `  M )   &    |-  H  =  ( K  .x.  .1.  )   &    |-  O  =  ( 1r `  A )   &    |-  .*  =  ( .s `  A )   &    |-  T  =  ( N matToPolyMat  R )   &    |-  W  =  (
 Base `  Y )   &    |-  Q  =  (Poly1 `  A )   &    |-  Z  =  (var1 `  A )   &    |-  .xb  =  ( .s `  Q )   &    |-  E  =  (.g `  (mulGrp `  Q ) )   &    |-  I  =  ( N pMatToMatPoly  R )   =>    |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B ) 
 ->  ( I `  H )  =  ( Q  gsumg  ( n  e.  NN0  |->  ( ( ( (coe1 `  K ) `  n )  .*  O ) 
 .xb  ( n E Z ) ) ) ) )
 
TheoremcpmadugsumlemB 19137* Lemma B for cpmadugsum 19141. (Contributed by AV, 2-Nov-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  P  =  (Poly1 `  R )   &    |-  Y  =  ( N Mat  P )   &    |-  T  =  ( N matToPolyMat  R )   &    |-  X  =  (var1 `  R )   &    |-  .^  =  (.g `  (mulGrp `  P ) )   &    |-  .x.  =  ( .s `  Y )   &    |-  .X. 
 =  ( .r `  Y )   &    |-  .1.  =  ( 1r `  Y )   =>    |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B ) 
 /\  ( s  e. 
 NN0  /\  b  e.  ( B  ^m  ( 0 ... s ) ) ) )  ->  ( ( X  .x.  .1.  )  .X.  ( Y  gsumg  ( i  e.  (
 0 ... s )  |->  ( ( i  .^  X )  .x.  ( T `  ( b `  i
 ) ) ) ) ) )  =  ( Y  gsumg  ( i  e.  (
 0 ... s )  |->  ( ( ( i  +  1 )  .^  X ) 
 .x.  ( T `  ( b `  i
 ) ) ) ) ) )
 
TheoremcpmadugsumlemC 19138* Lemma C for cpmadugsum 19141. (Contributed by AV, 2-Nov-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  P  =  (Poly1 `  R )   &    |-  Y  =  ( N Mat  P )   &    |-  T  =  ( N matToPolyMat  R )   &    |-  X  =  (var1 `  R )   &    |-  .^  =  (.g `  (mulGrp `  P ) )   &    |-  .x.  =  ( .s `  Y )   &    |-  .X. 
 =  ( .r `  Y )   &    |-  .1.  =  ( 1r `  Y )   =>    |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B ) 
 /\  ( s  e. 
 NN0  /\  b  e.  ( B  ^m  ( 0 ... s ) ) ) )  ->  ( ( T `  M )  .X.  ( Y  gsumg  ( i  e.  (
 0 ... s )  |->  ( ( i  .^  X )  .x.  ( T `  ( b `  i
 ) ) ) ) ) )  =  ( Y  gsumg  ( i  e.  (
 0 ... s )  |->  ( ( i  .^  X )  .x.  ( ( T `
  M )  .X.  ( T `  ( b `
  i ) ) ) ) ) ) )
 
TheoremcpmadugsumlemF 19139* Lemma F for cpmadugsum 19141. (Contributed by AV, 7-Nov-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  P  =  (Poly1 `  R )   &    |-  Y  =  ( N Mat  P )   &    |-  T  =  ( N matToPolyMat  R )   &    |-  X  =  (var1 `  R )   &    |-  .^  =  (.g `  (mulGrp `  P ) )   &    |-  .x.  =  ( .s `  Y )   &    |-  .X. 
 =  ( .r `  Y )   &    |-  .1.  =  ( 1r `  Y )   &    |-  .+  =  ( +g  `  Y )   &    |-  .-  =  ( -g `  Y )   =>    |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B ) 
 /\  ( s  e. 
 NN  /\  b  e.  ( B  ^m  ( 0
 ... s ) ) ) )  ->  (
 ( ( X  .x.  .1.  )  .X.  ( Y  gsumg  (
 i  e.  ( 0
 ... s )  |->  ( ( i  .^  X )  .x.  ( T `  ( b `  i
 ) ) ) ) ) )  .-  (
 ( T `  M )  .X.  ( Y  gsumg  ( i  e.  ( 0 ... s )  |->  ( ( i  .^  X )  .x.  ( T `  (
 b `  i )
 ) ) ) ) ) )  =  ( ( Y  gsumg  ( i  e.  (
 1 ... s )  |->  ( ( i  .^  X )  .x.  ( ( T `
  ( b `  ( i  -  1
 ) ) )  .-  ( ( T `  M )  .X.  ( T `
  ( b `  i ) ) ) ) ) ) ) 
 .+  ( ( ( ( s  +  1 )  .^  X )  .x.  ( T `  (
 b `  s )
 ) )  .-  (
 ( T `  M )  .X.  ( T `  ( b `  0
 ) ) ) ) ) )
 
Theoremcpmadugsumfi 19140* The product of the characteristic matrix of a given matrix and its adjunct represented as finite sum. (Contributed by AV, 7-Nov-2019.) (Proof shortened by AV, 29-Nov-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  P  =  (Poly1 `  R )   &    |-  Y  =  ( N Mat  P )   &    |-  T  =  ( N matToPolyMat  R )   &    |-  X  =  (var1 `  R )   &    |-  .^  =  (.g `  (mulGrp `  P ) )   &    |-  .x.  =  ( .s `  Y )   &    |-  .X. 
 =  ( .r `  Y )   &    |-  .1.  =  ( 1r `  Y )   &    |-  .+  =  ( +g  `  Y )   &    |-  .-  =  ( -g `  Y )   &    |-  I  =  ( ( X  .x.  .1.  )  .-  ( T `  M ) )   &    |-  J  =  ( N maAdju  P )   =>    |-  (
 ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  E. s  e.  NN  E. b  e.  ( B  ^m  (
 0 ... s ) ) ( I  .X.  ( J `  I ) )  =  ( ( Y 
 gsumg  ( i  e.  (
 1 ... s )  |->  ( ( i  .^  X )  .x.  ( ( T `
  ( b `  ( i  -  1
 ) ) )  .-  ( ( T `  M )  .X.  ( T `
  ( b `  i ) ) ) ) ) ) ) 
 .+  ( ( ( ( s  +  1 )  .^  X )  .x.  ( T `  (
 b `  s )
 ) )  .-  (
 ( T `  M )  .X.  ( T `  ( b `  0
 ) ) ) ) ) )
 
Theoremcpmadugsum 19141* The product of the characteristic matrix of a given matrix and its adjunct represented as an infinite sum. (Contributed by AV, 10-Nov-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  P  =  (Poly1 `  R )   &    |-  Y  =  ( N Mat  P )   &    |-  T  =  ( N matToPolyMat  R )   &    |-  X  =  (var1 `  R )   &    |-  .^  =  (.g `  (mulGrp `  P ) )   &    |-  .x.  =  ( .s `  Y )   &    |-  .X. 
 =  ( .r `  Y )   &    |-  .1.  =  ( 1r `  Y )   &    |-  .+  =  ( +g  `  Y )   &    |-  .-  =  ( -g `  Y )   &    |-  I  =  ( ( X  .x.  .1.  )  .-  ( T `  M ) )   &    |-  J  =  ( N maAdju  P )   &    |-  .0.  =  ( 0g `  Y )   &    |-  G  =  ( n  e.  NN0  |->  if ( n  =  0 ,  (  .0.  .-  ( ( T `  M )  .X.  ( T `  ( b `
  0 ) ) ) ) ,  if ( n  =  (
 s  +  1 ) ,  ( T `  ( b `  s
 ) ) ,  if ( ( s  +  1 )  <  n ,  .0.  ,  ( ( T `
  ( b `  ( n  -  1
 ) ) )  .-  ( ( T `  M )  .X.  ( T `
  ( b `  n ) ) ) ) ) ) ) )   =>    |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  E. s  e.  NN  E. b  e.  ( B  ^m  (
 0 ... s ) ) ( I  .X.  ( J `  I ) )  =  ( Y  gsumg  ( i  e.  NN0  |->  ( ( i  .^  X )  .x.  ( G `  i
 ) ) ) ) )
 
Theoremcpmidgsum2 19142* Representation of the identity matrix multiplied with the characteristic polynomial of a matrix as another group sum. (Contributed by AV, 10-Nov-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  P  =  (Poly1 `  R )   &    |-  Y  =  ( N Mat  P )   &    |-  T  =  ( N matToPolyMat  R )   &    |-  X  =  (var1 `  R )   &    |-  .^  =  (.g `  (mulGrp `  P ) )   &    |-  .x.  =  ( .s `  Y )   &    |-  .X. 
 =  ( .r `  Y )   &    |-  .1.  =  ( 1r `  Y )   &    |-  .+  =  ( +g  `  Y )   &    |-  .-  =  ( -g `  Y )   &    |-  I  =  ( ( X  .x.  .1.  )  .-  ( T `  M ) )   &    |-  J  =  ( N maAdju  P )   &    |-  .0.  =  ( 0g `  Y )   &    |-  G  =  ( n  e.  NN0  |->  if ( n  =  0 ,  (  .0.  .-  ( ( T `  M )  .X.  ( T `  ( b `
  0 ) ) ) ) ,  if ( n  =  (
 s  +  1 ) ,  ( T `  ( b `  s
 ) ) ,  if ( ( s  +  1 )  <  n ,  .0.  ,  ( ( T `
  ( b `  ( n  -  1
 ) ) )  .-  ( ( T `  M )  .X.  ( T `
  ( b `  n ) ) ) ) ) ) ) )   &    |-  C  =  ( N CharPlyMat  R )   &    |-  K  =  ( C `  M )   &    |-  H  =  ( K  .x.  .1.  )   =>    |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  E. s  e.  NN  E. b  e.  ( B  ^m  (
 0 ... s ) ) H  =  ( Y 
 gsumg  ( i  e.  NN0  |->  ( ( i  .^  X )  .x.  ( G `  i
 ) ) ) ) )
 
Theoremcpmidg2sum 19143* Equality of two sums representing the identity matrix multiplied with the characteristic polynomial of a matrix. (Contributed by AV, 11-Nov-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  P  =  (Poly1 `  R )   &    |-  Y  =  ( N Mat  P )   &    |-  T  =  ( N matToPolyMat  R )   &    |-  X  =  (var1 `  R )   &    |-  .^  =  (.g `  (mulGrp `  P ) )   &    |-  .x.  =  ( .s `  Y )   &    |-  .X. 
 =  ( .r `  Y )   &    |-  .1.  =  ( 1r `  Y )   &    |-  .+  =  ( +g  `  Y )   &    |-  .-  =  ( -g `  Y )   &    |-  I  =  ( ( X  .x.  .1.  )  .-  ( T `  M ) )   &    |-  J  =  ( N maAdju  P )   &    |-  .0.  =  ( 0g `  Y )   &    |-  G  =  ( n  e.  NN0  |->  if ( n  =  0 ,  (  .0.  .-  ( ( T `  M )  .X.  ( T `  ( b `
  0 ) ) ) ) ,  if ( n  =  (
 s  +  1 ) ,  ( T `  ( b `  s
 ) ) ,  if ( ( s  +  1 )  <  n ,  .0.  ,  ( ( T `
  ( b `  ( n  -  1
 ) ) )  .-  ( ( T `  M )  .X.  ( T `
  ( b `  n ) ) ) ) ) ) ) )   &    |-  C  =  ( N CharPlyMat  R )   &    |-  K  =  ( C `  M )   &    |-  U  =  (algSc `  P )   =>    |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  E. s  e.  NN  E. b  e.  ( B  ^m  (
 0 ... s ) ) ( Y  gsumg  ( i  e.  NN0  |->  ( ( i  .^  X )  .x.  ( ( U `  ( (coe1 `  K ) `  i
 ) )  .x.  .1.  ) ) ) )  =  ( Y  gsumg  ( i  e.  NN0  |->  ( ( i  .^  X )  .x.  ( G `  i
 ) ) ) ) )
 
Theoremcpmadumatpolylem1 19144* Lemma 1 for cpmadumatpoly 19146. (Contributed by AV, 20-Nov-2019.) (Revised by AV, 15-Dec-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  P  =  (Poly1 `  R )   &    |-  Y  =  ( N Mat  P )   &    |-  T  =  ( N matToPolyMat  R )   &    |- 
 .X.  =  ( .r `  Y )   &    |-  .-  =  ( -g `  Y )   &    |-  .0.  =  ( 0g `  Y )   &    |-  G  =  ( n  e.  NN0  |->  if ( n  =  0 ,  (  .0.  .-  ( ( T `  M )  .X.  ( T `  ( b `
  0 ) ) ) ) ,  if ( n  =  (
 s  +  1 ) ,  ( T `  ( b `  s
 ) ) ,  if ( ( s  +  1 )  <  n ,  .0.  ,  ( ( T `
  ( b `  ( n  -  1
 ) ) )  .-  ( ( T `  M )  .X.  ( T `
  ( b `  n ) ) ) ) ) ) ) )   &    |-  S  =  ( N ConstPolyMat  R )   &    |-  .x.  =  ( .s `  Y )   &    |-  .1.  =  ( 1r `  Y )   &    |-  Z  =  (var1 `  R )   &    |-  D  =  ( ( Z  .x.  .1.  )  .-  ( T `  M ) )   &    |-  J  =  ( N maAdju  P )   &    |-  W  =  (
 Base `  Y )   &    |-  Q  =  (Poly1 `  A )   &    |-  X  =  (var1 `  A )   &    |-  .*  =  ( .s `  Q )   &    |-  .^  =  (.g `  (mulGrp `  Q ) )   &    |-  U  =  ( N cPolyMatToMat  R )   =>    |-  ( ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  s  e.  NN )  /\  b  e.  ( B  ^m  (
 0 ... s ) ) )  ->  ( U  o.  G )  e.  ( B  ^m  NN0 ) )
 
Theoremcpmadumatpolylem2 19145* Lemma 2 for cpmadumatpoly 19146. (Contributed by AV, 20-Nov-2019.) (Revised by AV, 15-Dec-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  P  =  (Poly1 `  R )   &    |-  Y  =  ( N Mat  P )   &    |-  T  =  ( N matToPolyMat  R )   &    |- 
 .X.  =  ( .r `  Y )   &    |-  .-  =  ( -g `  Y )   &    |-  .0.  =  ( 0g `  Y )   &    |-  G  =  ( n  e.  NN0  |->  if ( n  =  0 ,  (  .0.  .-  ( ( T `  M )  .X.  ( T `  ( b `
  0 ) ) ) ) ,  if ( n  =  (
 s  +  1 ) ,  ( T `  ( b `  s
 ) ) ,  if ( ( s  +  1 )  <  n ,  .0.  ,  ( ( T `
  ( b `  ( n  -  1
 ) ) )  .-  ( ( T `  M )  .X.  ( T `
  ( b `  n ) ) ) ) ) ) ) )   &    |-  S  =  ( N ConstPolyMat  R )   &    |-  .x.  =  ( .s `  Y )   &    |-  .1.  =  ( 1r `  Y )   &    |-  Z  =  (var1 `  R )   &    |-  D  =  ( ( Z  .x.  .1.  )  .-  ( T `  M ) )   &    |-  J  =  ( N maAdju  P )   &    |-  W  =  (
 Base `  Y )   &    |-  Q  =  (Poly1 `  A )   &    |-  X  =  (var1 `  A )   &    |-  .*  =  ( .s `  Q )   &    |-  .^  =  (.g `  (mulGrp `  Q ) )   &    |-  U  =  ( N cPolyMatToMat  R )   =>    |-  ( ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  s  e.  NN )  /\  b  e.  ( B  ^m  (
 0 ... s ) ) )  ->  ( U  o.  G ) finSupp  ( 0g
 `  A ) )
 
Theoremcpmadumatpoly 19146* The product of the characteristic matrix of a given matrix and its adjunct represented as a polynomial over matrices. (Contributed by AV, 20-Nov-2019.) (Revised by AV, 7-Dec-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  P  =  (Poly1 `  R )   &    |-  Y  =  ( N Mat  P )   &    |-  T  =  ( N matToPolyMat  R )   &    |- 
 .X.  =  ( .r `  Y )   &    |-  .-  =  ( -g `  Y )   &    |-  .0.  =  ( 0g `  Y )   &    |-  G  =  ( n  e.  NN0  |->  if ( n  =  0 ,  (  .0.  .-  ( ( T `  M )  .X.  ( T `  ( b `
  0 ) ) ) ) ,  if ( n  =  (
 s  +  1 ) ,  ( T `  ( b `  s
 ) ) ,  if ( ( s  +  1 )  <  n ,  .0.  ,  ( ( T `
  ( b `  ( n  -  1
 ) ) )  .-  ( ( T `  M )  .X.  ( T `
  ( b `  n ) ) ) ) ) ) ) )   &    |-  S  =  ( N ConstPolyMat  R )   &    |-  .x.  =  ( .s `  Y )   &    |-  .1.  =  ( 1r `  Y )   &    |-  Z  =  (var1 `  R )   &    |-  D  =  ( ( Z  .x.  .1.  )  .-  ( T `  M ) )   &    |-  J  =  ( N maAdju  P )   &    |-  W  =  (
 Base `  Y )   &    |-  Q  =  (Poly1 `  A )   &    |-  X  =  (var1 `  A )   &    |-  .*  =  ( .s `  Q )   &    |-  .^  =  (.g `  (mulGrp `  Q ) )   &    |-  U  =  ( N cPolyMatToMat  R )   &    |-  I  =  ( N pMatToMatPoly  R )   =>    |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B ) 
 ->  E. s  e.  NN  E. b  e.  ( B 
 ^m  ( 0 ... s ) ) ( I `  ( D 
 .X.  ( J `  D ) ) )  =  ( Q  gsumg  ( n  e.  NN0  |->  ( ( U `  ( G `
  n ) )  .*  ( n  .^  X ) ) ) ) )
 
Theoremcayhamlem2 19147 Lemma for cayhamlem3 19150. (Contributed by AV, 24-Nov-2019.)
 |-  K  =  ( Base `  R )   &    |-  A  =  ( N Mat  R )   &    |-  B  =  ( Base `  A )   &    |-  .1.  =  ( 1r `  A )   &    |- 
 .*  =  ( .s
 `  A )   &    |-  .^  =  (.g `  (mulGrp `  A )
 )   &    |- 
 .x.  =  ( .r `  A )   =>    |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B ) 
 /\  ( H  e.  ( K  ^m  NN0 )  /\  L  e.  NN0 )
 )  ->  ( ( H `  L )  .*  ( L  .^  M ) )  =  ( ( L  .^  M )  .x.  ( ( H `  L )  .*  .1.  )
 ) )
 
Theoremchcoeffeqlem 19148* Lemma for chcoeffeq 19149. (Contributed by AV, 21-Nov-2019.) (Proof shortened by AV, 7-Dec-2019.) (Revised by AV, 15-Dec-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  P  =  (Poly1 `  R )   &    |-  Y  =  ( N Mat  P )   &    |-  .X. 
 =  ( .r `  Y )   &    |-  .-  =  ( -g `  Y )   &    |-  .0.  =  ( 0g `  Y )   &    |-  T  =  ( N matToPolyMat  R )   &    |-  C  =  ( N CharPlyMat  R )   &    |-  K  =  ( C `  M )   &    |-  G  =  ( n  e.  NN0  |->  if ( n  =  0 ,  (  .0.  .-  ( ( T `  M )  .X.  ( T `
  ( b `  0 ) ) ) ) ,  if ( n  =  ( s  +  1 ) ,  ( T `  (
 b `  s )
 ) ,  if (
 ( s  +  1 )  <  n ,  .0.  ,  ( ( T `
  ( b `  ( n  -  1
 ) ) )  .-  ( ( T `  M )  .X.  ( T `
  ( b `  n ) ) ) ) ) ) ) )   &    |-  W  =  (
 Base `  Y )   &    |-  .1.  =  ( 1r `  A )   &    |- 
 .*  =  ( .s
 `  A )   &    |-  U  =  ( N cPolyMatToMat  R )   =>    |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  /\  (
 s  e.  NN  /\  b  e.  ( B  ^m  ( 0 ... s
 ) ) ) ) 
 ->  ( ( (Poly1 `  A )  gsumg  ( n  e.  NN0  |->  ( ( U `  ( G `  n ) ) ( .s `  (Poly1 `  A ) ) ( n (.g `  (mulGrp `  (Poly1 `  A ) ) ) (var1 `  A ) ) ) ) )  =  ( (Poly1 `  A )  gsumg  ( n  e.  NN0  |->  ( ( ( (coe1 `  K ) `  n )  .*  .1.  )
 ( .s `  (Poly1 `  A ) ) ( n (.g `  (mulGrp `  (Poly1 `  A ) ) ) (var1 `  A ) ) ) ) )  ->  A. n  e.  NN0  ( U `  ( G `  n ) )  =  ( ( (coe1 `  K ) `  n )  .*  .1.  ) ) )
 
Theoremchcoeffeq 19149* The coefficients of the characteristic polynomial multiplied with the identity matrix represented by (transformed) ring elements obtained from the adjunct of the characteristic matrix. (Contributed by AV, 21-Nov-2019.) (Proof shortened by AV, 8-Dec-2019.) (Revised by AV, 15-Dec-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  P  =  (Poly1 `  R )   &    |-  Y  =  ( N Mat  P )   &    |-  .X. 
 =  ( .r `  Y )   &    |-  .-  =  ( -g `  Y )   &    |-  .0.  =  ( 0g `  Y )   &    |-  T  =  ( N matToPolyMat  R )   &    |-  C  =  ( N CharPlyMat  R )   &    |-  K  =  ( C `  M )   &    |-  G  =  ( n  e.  NN0  |->  if ( n  =  0 ,  (  .0.  .-  ( ( T `  M )  .X.  ( T `
  ( b `  0 ) ) ) ) ,  if ( n  =  ( s  +  1 ) ,  ( T `  (
 b `  s )
 ) ,  if (
 ( s  +  1 )  <  n ,  .0.  ,  ( ( T `
  ( b `  ( n  -  1
 ) ) )  .-  ( ( T `  M )  .X.  ( T `
  ( b `  n ) ) ) ) ) ) ) )   &    |-  W  =  (
 Base `  Y )   &    |-  .1.  =  ( 1r `  A )   &    |- 
 .*  =  ( .s
 `  A )   &    |-  U  =  ( N cPolyMatToMat  R )   =>    |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B ) 
 ->  E. s  e.  NN  E. b  e.  ( B 
 ^m  ( 0 ... s ) ) A. n  e.  NN0  ( U `
  ( G `  n ) )  =  ( ( (coe1 `  K ) `  n )  .*  .1.  ) )
 
Theoremcayhamlem3 19150* Lemma for cayhamlem4 19151. (Contributed by AV, 24-Nov-2019.) (Revised by AV, 15-Dec-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  P  =  (Poly1 `  R )   &    |-  Y  =  ( N Mat  P )   &    |-  .X. 
 =  ( .r `  Y )   &    |-  .-  =  ( -g `  Y )   &    |-  .0.  =  ( 0g `  Y )   &    |-  T  =  ( N matToPolyMat  R )   &    |-  C  =  ( N CharPlyMat  R )   &    |-  K  =  ( C `  M )   &    |-  G  =  ( n  e.  NN0  |->  if ( n  =  0 ,  (  .0.  .-  ( ( T `  M )  .X.  ( T `
  ( b `  0 ) ) ) ) ,  if ( n  =  ( s  +  1 ) ,  ( T `  (
 b `  s )
 ) ,  if (
 ( s  +  1 )  <  n ,  .0.  ,  ( ( T `
  ( b `  ( n  -  1
 ) ) )  .-  ( ( T `  M )  .X.  ( T `
  ( b `  n ) ) ) ) ) ) ) )   &    |-  W  =  (
 Base `  Y )   &    |-  .1.  =  ( 1r `  A )   &    |- 
 .*  =  ( .s
 `  A )   &    |-  U  =  ( N cPolyMatToMat  R )   &    |-  .^  =  (.g `  (mulGrp `  A )
 )   &    |- 
 .x.  =  ( .r `  A )   =>    |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  E. s  e.  NN  E. b  e.  ( B  ^m  (
 0 ... s ) ) ( A  gsumg  ( n  e.  NN0  |->  ( ( (coe1 `  K ) `  n )  .*  ( n  .^  M ) ) ) )  =  ( A  gsumg  ( n  e.  NN0  |->  ( ( n  .^  M )  .x.  ( U `
  ( G `  n ) ) ) ) ) )
 
Theoremcayhamlem4 19151* Lemma for cayleyhamilton 19153. (Contributed by AV, 25-Nov-2019.) (Revised by AV, 15-Dec-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  P  =  (Poly1 `  R )   &    |-  Y  =  ( N Mat  P )   &    |-  .X. 
 =  ( .r `  Y )   &    |-  .-  =  ( -g `  Y )   &    |-  .0.  =  ( 0g `  Y )   &    |-  T  =  ( N matToPolyMat  R )   &    |-  C  =  ( N CharPlyMat  R )   &    |-  K  =  ( C `  M )   &    |-  G  =  ( n  e.  NN0  |->  if ( n  =  0 ,  (  .0.  .-  ( ( T `  M )  .X.  ( T `
  ( b `  0 ) ) ) ) ,  if ( n  =  ( s  +  1 ) ,  ( T `  (
 b `  s )
 ) ,  if (
 ( s  +  1 )  <  n ,  .0.  ,  ( ( T `
  ( b `  ( n  -  1
 ) ) )  .-  ( ( T `  M )  .X.  ( T `
  ( b `  n ) ) ) ) ) ) ) )   &    |-  W  =  (
 Base `  Y )   &    |-  .1.  =  ( 1r `  A )   &    |- 
 .*  =  ( .s
 `  A )   &    |-  U  =  ( N cPolyMatToMat  R )   &    |-  .^  =  (.g `  (mulGrp `  A )
 )   &    |-  E  =  (.g `  (mulGrp `  Y ) )   =>    |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B ) 
 ->  E. s  e.  NN  E. b  e.  ( B 
 ^m  ( 0 ... s ) ) ( A  gsumg  ( n  e.  NN0  |->  ( ( (coe1 `  K ) `  n )  .*  ( n  .^  M ) ) ) )  =  ( U `  ( Y  gsumg  ( n  e.  NN0  |->  ( ( n E ( T `  M ) )  .X.  ( G `
  n ) ) ) ) ) )
 
Theoremcayleyhamilton0 19152* The Cayley-Hamilton theorem: A matrix over a commutative ring "satisfies its own characteristic equation". This version of cayleyhamilton 19153 provides definitions not used in the theorem itself, but in its proof to make it clearer, more readable and shorter compared with a proof without them (see cayleyhamiltonALT 19154)! (Contributed by AV, 25-Nov-2019.) (Revised by AV, 15-Dec-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  .0.  =  ( 0g `  A )   &    |- 
 .1.  =  ( 1r `  A )   &    |-  .*  =  ( .s `  A )   &    |-  .^  =  (.g `  (mulGrp `  A ) )   &    |-  C  =  ( N CharPlyMat  R )   &    |-  K  =  (coe1 `  ( C `  M ) )   &    |-  P  =  (Poly1 `  R )   &    |-  Y  =  ( N Mat  P )   &    |-  .X.  =  ( .r `  Y )   &    |-  .-  =  ( -g `  Y )   &    |-  Z  =  ( 0g
 `  Y )   &    |-  W  =  ( Base `  Y )   &    |-  E  =  (.g `  (mulGrp `  Y ) )   &    |-  T  =  ( N matToPolyMat  R )   &    |-  G  =  ( n  e.  NN0  |->  if ( n  =  0 ,  ( Z  .-  ( ( T `  M ) 
 .X.  ( T `  ( b `  0
 ) ) ) ) ,  if ( n  =  ( s  +  1 ) ,  ( T `  ( b `  s ) ) ,  if ( ( s  +  1 )  < 
 n ,  Z ,  ( ( T `  ( b `  ( n  -  1 ) ) )  .-  ( ( T `  M )  .X.  ( T `  ( b `
  n ) ) ) ) ) ) ) )   &    |-  U  =  ( N cPolyMatToMat  R )   =>    |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  ( A  gsumg  ( n  e.  NN0  |->  ( ( K `  n )  .*  ( n  .^  M ) ) ) )  =  .0.  )
 
Theoremcayleyhamilton 19153* The Cayley-Hamilton theorem: A matrix over a commutative ring "satisfies its own characteristic equation", see theorem 7.8 in [Roman] p. 170 (without proof!), or theorem 3.1 in [Lang] p. 561. In other words, a matrix over a commutative ring "inserted" into its characteristic polynomial results in zero. This is Metamath 100 proof #49. (Contributed by Alexander van der Vekens, 25-Nov-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  .0.  =  ( 0g `  A )   &    |-  C  =  ( N CharPlyMat  R )   &    |-  K  =  (coe1 `  ( C `  M ) )   &    |-  .*  =  ( .s `  A )   &    |-  .^  =  (.g `  (mulGrp `  A ) )   =>    |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  ( A  gsumg  ( n  e.  NN0  |->  ( ( K `  n )  .*  ( n  .^  M ) ) ) )  =  .0.  )
 
TheoremcayleyhamiltonALT 19154* Alternate proof of cayleyhamilton 19153, the Cayley-Hamilton theorem. This proof does not use cayleyhamilton0 19152 directly, but has the same structure as the proof of cayleyhamilton0 19152. In contrast to the proof of cayleyhamilton0 19152, only the definitions required to formulate the theorem itself are used, causing the definitions used in the lemmata being expanded, which makes the proof longer and more difficult to read. (Contributed by AV, 25-Nov-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  .0.  =  ( 0g `  A )   &    |-  C  =  ( N CharPlyMat  R )   &    |-  K  =  (coe1 `  ( C `  M ) )   &    |-  .*  =  ( .s `  A )   &    |-  .^  =  (.g `  (mulGrp `  A ) )   =>    |-  ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B )  ->  ( A  gsumg  ( n  e.  NN0  |->  ( ( K `  n )  .*  ( n  .^  M ) ) ) )  =  .0.  )
 
Theoremcayleyhamilton1 19155* The Cayley-Hamilton theorem: A matrix over a commutative ring "satisfies its own characteristic equation", or, in other words, a matrix over a commutative ring "inserted" into its characteristic polynomial results in zero. In this variant of cayleyhamilton 19153, the meaning of "inserted" is made more transparent: If the characteristic polynomial is a polynomial with coefficients  ( F `  n ), then a matrix over a commutative ring "inserted" into its characteristic polynomial is the sum of these coefficients multiplied with the corresponding power of the matrix. (Contributed by AV, 25-Nov-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  .0.  =  ( 0g `  A )   &    |-  C  =  ( N CharPlyMat  R )   &    |-  K  =  (coe1 `  ( C `  M ) )   &    |-  .*  =  ( .s `  A )   &    |-  .^  =  (.g `  (mulGrp `  A ) )   &    |-  L  =  (
 Base `  R )   &    |-  X  =  (var1 `  R )   &    |-  P  =  (Poly1 `  R )   &    |-  .x.  =  ( .s `  P )   &    |-  E  =  (.g `  (mulGrp `  P ) )   &    |-  Z  =  ( 0g `  R )   =>    |-  ( ( ( N  e.  Fin  /\  R  e.  CRing  /\  M  e.  B ) 
 /\  ( F  e.  ( L  ^m  NN0 )  /\  F finSupp  Z ) )  ->  ( ( C `  M )  =  ( P  gsumg  ( n  e.  NN0  |->  ( ( F `  n )  .x.  ( n E X ) ) ) )  ->  ( A  gsumg  ( n  e.  NN0  |->  ( ( F `  n )  .*  ( n  .^  M ) ) ) )  =  .0.  ) )
 
PART 12  BASIC TOPOLOGY
 
12.1  Topology
 
12.1.1  Topological spaces
 
Syntaxctop 19156 Extend class notation with the class of all topologies.
 class  Top
 
Syntaxctopon 19157 The class function of all topologies over a base set.
 class TopOn
 
SyntaxctpsOLD 19158 Extend class notation with the class of all topological spaces. (New usage is discouraged.)
 class  TopSpOLD
 
Syntaxctps 19159 Extend class notation with the class of all topological spaces.
 class  TopSp
 
Syntaxctb 19160 Extend class notation with the class of all topological bases.
 class  TopBases
 
Definitiondf-top 19161* Define the (proper) class of all topologies. See istop2g 19167 for an alternate way to express finite intersection and istps5OLD 19187 for a standard definition as an ordered pair of a set and a topology on it.

The final form of the definition is due to Bourbaki (Def. 1 of [BourbakiTop1] p. I.1), while the idea of defining a topology in terms of its open sets is due to Aleksandrov. For the convoluted history of the definitions of these notions, see

Gregory H. Moore, The emergence of open sets, closed sets, and limit points in analysis and topology, Historia Mathematica 35 (2008) 220--241.

(Contributed by NM, 3-Mar-2006.) (Revised by BJ, 20-Oct-2018.)

 |- 
 Top  =  { x  |  ( A. y  e. 
 ~P  x U. y  e.  x  /\  A. y  e.  x  A. z  e.  x  ( y  i^i  z )  e.  x ) }
 
Definitiondf-topspOLD 19162* Define the class of all topological spaces, each of which is an ordered pair the second of which is a topology on the first. See istps5OLD 19187 for a standard way to express a topological space. (Contributed by NM, 8-Mar-2006.) (New usage is discouraged.)
 |- 
 TopSpOLD  =  { <. x ,  y >.  |  ( y  e.  Top  /\  x  =  U. y ) }
 
Definitiondf-bases 19163* Define the class of all topological bases. Equivalent to definition of basis in [Munkres] p. 78 (see isbasis2g 19211). Note that "bases" is the plural of "basis." (Contributed by NM, 17-Jul-2006.)
 |-  TopBases 
 =  { x  |  A. y  e.  x  A. z  e.  x  ( y  i^i  z ) 
 C_  U. ( x  i^i  ~P ( y  i^i  z
 ) ) }
 
Definitiondf-topon 19164* Define the set of topologies with a given base set. (Contributed by Stefan O'Rear, 31-Jan-2015.)
 |- TopOn  =  ( b  e.  _V  |->  { j  e.  Top  |  b  =  U. j }
 )
 
Definitiondf-topsp 19165 Define the class of all topological spaces (structures). (Contributed by Stefan O'Rear, 13-Aug-2015.)
 |- 
 TopSp  =  { f  |  ( TopOpen `  f )  e.  (TopOn `  ( Base `  f ) ) }
 
Theoremistopg 19166* Express the predicate " J is a topology." Note: In the literature, a topology is often represented by a script letter T, which resembles the letter J. This confusion may have led to J being used by some authors - e.g. K. D. Joshi, Introduction to General Topology (1983), p. 114 - and it is convenient for us since we later use  T to represent linear transformations (operators). (Contributed by Stefan Allan, 3-Mar-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
 |-  ( J  e.  A  ->  ( J  e.  Top  <->  ( A. x ( x  C_  J  ->  U. x  e.  J )  /\  A. x  e.  J  A. y  e.  J  ( x  i^i  y )  e.  J ) ) )
 
Theoremistop2g 19167* Express the predicate " J is a topology," using "the intersection of the elements of any finite subcollection" instead of the intersection of any two elements. (Contributed by NM, 19-Jul-2006.)
 |-  ( J  e.  A  ->  ( J  e.  Top  <->  ( A. x ( x  C_  J  ->  U. x  e.  J )  /\  A. x ( ( x  C_  J  /\  x  =/=  (/)  /\  x  e.  Fin )  ->  |^| x  e.  J ) ) ) )
 
Theoremuniopn 19168 The union of a subset of a topology is an open set. (Contributed by Stefan Allan, 27-Feb-2006.)
 |-  ( ( J  e.  Top  /\  A  C_  J )  ->  U. A  e.  J )
 
Theoremiunopn 19169* The indexed union of a subset of a topology is an open set. (Contributed by NM, 5-Oct-2006.)
 |-  ( ( J  e.  Top  /\  A. x  e.  A  B  e.  J )  -> 
 U_ x  e.  A  B  e.  J )
 
Theoreminopn 19170 The intersection of two open sets of a topology is also an open set. (Contributed by NM, 17-Jul-2006.)
 |-  ( ( J  e.  Top  /\  A  e.  J  /\  B  e.  J )  ->  ( A  i^i  B )  e.  J )
 
Theoremfitop 19171 A topology is closed under finite intersections. (Contributed by Jeff Hankins, 7-Oct-2009.)
 |-  ( J  e.  Top  ->  ( fi `  J )  =  J )
 
Theoremfiinopn 19172 The intersection of a nonempty finite family of open sets is open. (Contributed by FL, 20-Apr-2012.)
 |-  ( J  e.  Top  ->  ( ( A  C_  J  /\  A  =/=  (/)  /\  A  e.  Fin )  ->  |^| A  e.  J ) )
 
Theoremiinopn 19173* The intersection of a nonempty finite family of open sets is open. (Contributed by Mario Carneiro, 14-Sep-2014.)
 |-  ( ( J  e.  Top  /\  ( A  e.  Fin  /\  A  =/=  (/)  /\  A. x  e.  A  B  e.  J ) )  ->  |^|_
 x  e.  A  B  e.  J )
 
Theoremunopn 19174 The union of two open sets is open. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ( J  e.  Top  /\  A  e.  J  /\  B  e.  J )  ->  ( A  u.  B )  e.  J )
 
Theorem0opn 19175 The empty set is an open subset of a topology. (Contributed by Stefan Allan, 27-Feb-2006.)
 |-  ( J  e.  Top  ->  (/) 
 e.  J )
 
Theorem0ntop 19176 The empty set is not a topology. (Contributed by FL, 1-Jun-2008.)
 |- 
 -.  (/)  e.  Top
 
Theoremtopopn 19177 The underlying set of a topology is an open set. (Contributed by NM, 17-Jul-2006.)
 |-  X  =  U. J   =>    |-  ( J  e.  Top  ->  X  e.  J )
 
Theoremeltopss 19178 A member of a topology is a subset of its underlying set. (Contributed by NM, 12-Sep-2006.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  A  e.  J ) 
 ->  A  C_  X )
 
Theoremriinopn 19179* A finite indexed relative intersection of open sets is open. (Contributed by Mario Carneiro, 22-Aug-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  A  e.  Fin  /\  A. x  e.  A  B  e.  J )  ->  ( X  i^i  |^|_ x  e.  A  B )  e.  J )
 
Theoremrintopn 19180 A finite relative intersection of open sets is open. (Contributed by Mario Carneiro, 22-Aug-2015.)
 |-  X  =  U. J   =>    |-  (
 ( J  e.  Top  /\  A  C_  J  /\  A  e.  Fin )  ->  ( X  i^i  |^| A )  e.  J )
 
TheoremeltopspOLD 19181 Construct a topological space from a topology and vice-versa. We say that  A is a topology on  U. A. (This could be proved more efficiently from istpsOLD 19183, but the proof here does not require the Axiom of Regularity.) (Contributed by NM, 8-Mar-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( <. U. J ,  J >.  e.  TopSpOLD  <->  J  e.  Top )
 
TheoremtpsexOLD 19182 Existence implied by membership in a topological space. (Contributed by NM, 18-Jul-2006.) (Revised by Mario Carneiro, 30-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( <. A ,  J >.  e.  TopSpOLD  ->  ( A  e.  _V  /\  J  e.  _V ) )
 
TheoremistpsOLD 19183 Express the predicate "is a topological space." (Contributed by NM, 18-Jul-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( <. A ,  J >.  e.  TopSpOLD  <->  ( J  e.  Top  /\  A  =  U. J ) )
 
Theoremistps2OLD 19184 Express the predicate "is a topological space." (Contributed by NM, 18-Jul-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( <. A ,  J >.  e.  TopSpOLD  <->  ( ( J  e.  Top  /\  J  C_  ~P A )  /\  ( (/) 
 e.  J  /\  A  e.  J ) ) )
 
Theoremistps3OLD 19185* A standard textbook definition of a topological space. (Contributed by NM, 18-Jul-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( <. A ,  J >.  e.  TopSpOLD  <->  ( ( J 
 C_  ~P A  /\  (/)  e.  J  /\  A  e.  J ) 
 /\  ( A. x ( x  C_  J  ->  U. x  e.  J ) 
 /\  A. x  e.  J  A. y  e.  J  ( x  i^i  y )  e.  J ) ) )
 
Theoremistps4OLD 19186* A standard textbook definition of a topological space. (Contributed by NM, 19-Jul-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( <. A ,  J >.  e.  TopSpOLD  <->  ( ( J 
 C_  ~P A  /\  (/)  e.  J  /\  A  e.  J ) 
 /\  ( A. x ( x  C_  J  ->  U. x  e.  J ) 
 /\  A. x ( ( x  C_  J  /\  x  =/=  (/)  /\  x  e.  Fin )  ->  |^| x  e.  J ) ) ) )
 
Theoremistps5OLD 19187* A standard textbook definition of a topological space  <. A ,  J >.: a topology on  A is a collection  J of subsets of  A such that  (/) and  A are in  J and that is closed under union and finite intersection. Definition of topological space in [Munkres] p. 76. (Contributed by NM, 19-Jul-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( <. A ,  J >.  e.  TopSpOLD  <->  ( ( A. x  e.  J  x  C_  A  /\  (/)  e.  J  /\  A  e.  J ) 
 /\  ( A. x ( x  C_  J  ->  U. x  e.  J ) 
 /\  A. x ( ( x  C_  J  /\  x  =/=  (/)  /\  x  e.  Fin )  ->  |^| x  e.  J ) ) ) )
 
Theoremistopon 19188 Property of being a topology with a given base set. (Contributed by Stefan O'Rear, 31-Jan-2015.) (Revised by Mario Carneiro, 13-Aug-2015.)
 |-  ( J  e.  (TopOn `  B )  <->  ( J  e.  Top  /\  B  =  U. J ) )
 
Theoremtopontop 19189 A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  ( J  e.  (TopOn `  B )  ->  J  e.  Top )
 
Theoremtoponuni 19190 The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  ( J  e.  (TopOn `  B )  ->  B  =  U. J )
 
Theoremtoponmax 19191 The base set of a topology is an open set. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  ( J  e.  (TopOn `  B )  ->  B  e.  J )
 
Theoremtoponss 19192 A member of a topology is a subset of its underlying set. (Contributed by Mario Carneiro, 21-Aug-2015.)
 |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  J )  ->  A  C_  X )
 
Theoremtoponcom 19193 If  K is a topology on the base set of topology  J, then  J is a topology on the base of  K. (Contributed by Mario Carneiro, 22-Aug-2015.)
 |-  ( ( J  e.  Top  /\  K  e.  (TopOn `  U. J ) )  ->  J  e.  (TopOn `  U. K ) )
 
Theoremtopontopi 19194 A topology on a given base set is a topology. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  J  e.  (TopOn `  B )   =>    |-  J  e.  Top
 
Theoremtoponunii 19195 The base set of a topology on a given base set. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  J  e.  (TopOn `  B )   =>    |-  B  =  U. J
 
Theoremtoptopon 19196 Alternative definition of  Top in terms of TopOn. (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  X  =  U. J   =>    |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
 
Theoremtopgele 19197 The topologies over the same set have the greatest element (the discrete topology) and the least element (the indiscrete topology). (Contributed by FL, 18-Apr-2010.) (Revised by Mario Carneiro, 16-Sep-2015.)
 |-  ( J  e.  (TopOn `  X )  ->  ( { (/) ,  X }  C_  J  /\  J  C_  ~P X ) )
 
Theoremtopsn 19198 The only topology on a singleton is the discrete topology (which is also the indiscrete topology by pwsn 4234). (Contributed by FL, 5-Jan-2009.) (Revised by Mario Carneiro, 16-Sep-2015.)
 |-  ( J  e.  (TopOn ` 
 { A } )  ->  J  =  ~P { A } )
 
Theoremistps 19199 Express the predicate "is a topological space." (Contributed by Mario Carneiro, 13-Aug-2015.)
 |-  A  =  ( Base `  K )   &    |-  J  =  (
 TopOpen `  K )   =>    |-  ( K  e.  TopSp  <->  J  e.  (TopOn `  A ) )
 
Theoremistps2 19200 Express the predicate "is a topological space." (Contributed by NM, 20-Oct-2012.)
 |-  A  =  ( Base `  K )   &    |-  J  =  (
 TopOpen `  K )   =>    |-  ( K  e.  TopSp  <->  ( J  e.  Top  /\  A  =  U. J ) )
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144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800
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