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Theorem List for Metamath Proof Explorer - 19501-19600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremgsumcom3 19501* A commutative law for finitely supported iterated sums. (Contributed by Stefan O'Rear, 2-Nov-2015.)
 |-  B  =  ( Base `  G )   &    |-  .0.  =  ( 0g `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  C  e.  W )   &    |-  (
 ( ph  /\  ( j  e.  A  /\  k  e.  C ) )  ->  X  e.  B )   &    |-  ( ph  ->  U  e.  Fin )   &    |-  ( ( ph  /\  (
 ( j  e.  A  /\  k  e.  C )  /\  -.  j U k ) )  ->  X  =  .0.  )   =>    |-  ( ph  ->  ( G  gsumg  ( j  e.  A  |->  ( G 
 gsumg  ( k  e.  C  |->  X ) ) ) )  =  ( G 
 gsumg  ( k  e.  C  |->  ( G  gsumg  ( j  e.  A  |->  X ) ) ) ) )
 
Theoremgsumcom3fi 19502* A commutative law for finite iterated sums. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  B  =  ( Base `  G )   &    |-  ( ph  ->  G  e. CMnd )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  C  e.  Fin )   &    |-  ( ( ph  /\  (
 j  e.  A  /\  k  e.  C )
 )  ->  X  e.  B )   =>    |-  ( ph  ->  ( G  gsumg  ( j  e.  A  |->  ( G  gsumg  ( k  e.  C  |->  X ) ) ) )  =  ( G 
 gsumg  ( k  e.  C  |->  ( G  gsumg  ( j  e.  A  |->  X ) ) ) ) )
 
Theoremmamucl 19503 Operation closure of matrix multiplication. (Contributed by Stefan O'Rear, 2-Sep-2015.) (Proof shortened by AV, 23-Jul-2019.)
 |-  B  =  ( Base `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  F  =  ( R maMul  <. M ,  N ,  P >. )   &    |-  ( ph  ->  M  e.  Fin )   &    |-  ( ph  ->  N  e.  Fin )   &    |-  ( ph  ->  P  e.  Fin )   &    |-  ( ph  ->  X  e.  ( B  ^m  ( M  X.  N ) ) )   &    |-  ( ph  ->  Y  e.  ( B  ^m  ( N  X.  P ) ) )   =>    |-  ( ph  ->  ( X F Y )  e.  ( B  ^m  ( M  X.  P ) ) )
 
Theoremmamuass 19504 Matrix multiplication is associative, see also statement in [Lang] p. 505. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  B  =  ( Base `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  M  e.  Fin )   &    |-  ( ph  ->  N  e.  Fin )   &    |-  ( ph  ->  O  e.  Fin )   &    |-  ( ph  ->  P  e.  Fin )   &    |-  ( ph  ->  X  e.  ( B  ^m  ( M  X.  N ) ) )   &    |-  ( ph  ->  Y  e.  ( B  ^m  ( N  X.  O ) ) )   &    |-  ( ph  ->  Z  e.  ( B  ^m  ( O  X.  P ) ) )   &    |-  F  =  ( R maMul  <. M ,  N ,  O >. )   &    |-  G  =  ( R maMul  <. M ,  O ,  P >. )   &    |-  H  =  ( R maMul  <. M ,  N ,  P >. )   &    |-  I  =  ( R maMul  <. N ,  O ,  P >. )   =>    |-  ( ph  ->  (
 ( X F Y ) G Z )  =  ( X H ( Y I Z ) ) )
 
Theoremmamudi 19505 Matrix multiplication distributes over addition on the left. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 23-Jul-2019.)
 |-  B  =  ( Base `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  F  =  ( R maMul  <. M ,  N ,  O >. )   &    |-  ( ph  ->  M  e.  Fin )   &    |-  ( ph  ->  N  e.  Fin )   &    |-  ( ph  ->  O  e.  Fin )   &    |-  .+  =  ( +g  `  R )   &    |-  ( ph  ->  X  e.  ( B  ^m  ( M  X.  N ) ) )   &    |-  ( ph  ->  Y  e.  ( B  ^m  ( M  X.  N ) ) )   &    |-  ( ph  ->  Z  e.  ( B  ^m  ( N  X.  O ) ) )   =>    |-  ( ph  ->  (
 ( X  oF  .+  Y ) F Z )  =  ( ( X F Z )  oF  .+  ( Y F Z ) ) )
 
Theoremmamudir 19506 Matrix multiplication distributes over addition on the right. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 23-Jul-2019.)
 |-  B  =  ( Base `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  F  =  ( R maMul  <. M ,  N ,  O >. )   &    |-  ( ph  ->  M  e.  Fin )   &    |-  ( ph  ->  N  e.  Fin )   &    |-  ( ph  ->  O  e.  Fin )   &    |-  .+  =  ( +g  `  R )   &    |-  ( ph  ->  X  e.  ( B  ^m  ( M  X.  N ) ) )   &    |-  ( ph  ->  Y  e.  ( B  ^m  ( N  X.  O ) ) )   &    |-  ( ph  ->  Z  e.  ( B  ^m  ( N  X.  O ) ) )   =>    |-  ( ph  ->  ( X F ( Y  oF  .+  Z ) )  =  ( ( X F Y )  oF  .+  ( X F Z ) ) )
 
Theoremmamuvs1 19507 Matrix multiplication distributes over scalar multiplication on the left. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  B  =  ( Base `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  F  =  ( R maMul  <. M ,  N ,  O >. )   &    |-  ( ph  ->  M  e.  Fin )   &    |-  ( ph  ->  N  e.  Fin )   &    |-  ( ph  ->  O  e.  Fin )   &    |-  .x.  =  ( .r `  R )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  ( B  ^m  ( M  X.  N ) ) )   &    |-  ( ph  ->  Z  e.  ( B  ^m  ( N  X.  O ) ) )   =>    |-  ( ph  ->  (
 ( ( ( M  X.  N )  X.  { X } )  oF  .x.  Y ) F Z )  =  ( ( ( M  X.  O )  X.  { X } )  oF  .x.  ( Y F Z ) ) )
 
Theoremmamuvs2 19508 Matrix multiplication distributes over scalar multiplication on the left. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 22-Jul-2019.)
 |-  ( ph  ->  R  e.  CRing )   &    |-  B  =  (
 Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  F  =  ( R maMul  <. M ,  N ,  O >. )   &    |-  ( ph  ->  M  e.  Fin )   &    |-  ( ph  ->  N  e.  Fin )   &    |-  ( ph  ->  O  e.  Fin )   &    |-  ( ph  ->  X  e.  ( B  ^m  ( M  X.  N ) ) )   &    |-  ( ph  ->  Y  e.  B )   &    |-  ( ph  ->  Z  e.  ( B  ^m  ( N  X.  O ) ) )   =>    |-  ( ph  ->  ( X F ( ( ( N  X.  O )  X.  { Y }
 )  oF  .x.  Z ) )  =  ( ( ( M  X.  O )  X.  { Y } )  oF  .x.  ( X F Z ) ) )
 
11.2.2  Square matrices

In the following, the square matrix algebra is defined as extensible structure Mat. In this subsection, however, only square matrices and their basic properties are regarded. This includes showing that  ( N Mat  R ) is a left module, see matlmod 19531. That  ( N Mat  R ) is a ring and an associative algebra is shown in the next subsection, after theorems about the identity matrix are available. Nevertheless,  ( N Mat  R ) is called "matrix ring" or "matrix algebra" already in this subsection.

 
Syntaxcmat 19509 Syntax for the square matrix algebra.
 class Mat
 
Definitiondf-mat 19510* Define the algebra of n x n matrices over a ring r. (Contributed by Stefan O'Rear, 31-Aug-2015.)
 |- Mat 
 =  ( n  e. 
 Fin ,  r  e.  _V 
 |->  ( ( r freeLMod  ( n  X.  n ) ) sSet  <. ( .r `  ndx ) ,  ( r maMul  <. n ,  n ,  n >. ) >. ) )
 
Theoremmatbas0pc 19511 There is no matrix with a proper class either as dimension or as underlying ring. (Contributed by AV, 28-Dec-2018.)
 |-  ( -.  ( N  e.  _V  /\  R  e.  _V )  ->  ( Base `  ( N Mat  R ) )  =  (/) )
 
Theoremmatbas0 19512 There is no matrix for a not finite dimension or a proper class as the underlying ring. (Contributed by AV, 28-Dec-2018.)
 |-  ( -.  ( N  e.  Fin  /\  R  e.  _V )  ->  ( Base `  ( N Mat  R ) )  =  (/) )
 
Theoremmatval 19513 Value of the matrix algebra. (Contributed by Stefan O'Rear, 4-Sep-2015.)
 |-  A  =  ( N Mat 
 R )   &    |-  G  =  ( R freeLMod  ( N  X.  N ) )   &    |-  .x.  =  ( R maMul  <. N ,  N ,  N >. )   =>    |-  ( ( N  e.  Fin  /\  R  e.  V ) 
 ->  A  =  ( G sSet  <. ( .r `  ndx ) ,  .x.  >. ) )
 
Theoremmatrcl 19514 Reverse closure for the matrix algebra. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   =>    |-  ( X  e.  B  ->  ( N  e.  Fin  /\  R  e.  _V )
 )
 
Theoremmatbas 19515 The matrix ring has the same base set as its underlying group. (Contributed by Stefan O'Rear, 4-Sep-2015.)
 |-  A  =  ( N Mat 
 R )   &    |-  G  =  ( R freeLMod  ( N  X.  N ) )   =>    |-  ( ( N  e.  Fin  /\  R  e.  V ) 
 ->  ( Base `  G )  =  ( Base `  A )
 )
 
Theoremmatplusg 19516 The matrix ring has the same addition as its underlying group. (Contributed by Stefan O'Rear, 4-Sep-2015.)
 |-  A  =  ( N Mat 
 R )   &    |-  G  =  ( R freeLMod  ( N  X.  N ) )   =>    |-  ( ( N  e.  Fin  /\  R  e.  V ) 
 ->  ( +g  `  G )  =  ( +g  `  A ) )
 
Theoremmatsca 19517 The matrix ring has the same scalars as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.)
 |-  A  =  ( N Mat 
 R )   &    |-  G  =  ( R freeLMod  ( N  X.  N ) )   =>    |-  ( ( N  e.  Fin  /\  R  e.  V ) 
 ->  (Scalar `  G )  =  (Scalar `  A )
 )
 
Theoremmatvsca 19518 The matrix ring has the same scalar multiplication as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.)
 |-  A  =  ( N Mat 
 R )   &    |-  G  =  ( R freeLMod  ( N  X.  N ) )   =>    |-  ( ( N  e.  Fin  /\  R  e.  V ) 
 ->  ( .s `  G )  =  ( .s `  A ) )
 
Theoremmat0 19519 The matrix ring has the same zero as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.)
 |-  A  =  ( N Mat 
 R )   &    |-  G  =  ( R freeLMod  ( N  X.  N ) )   =>    |-  ( ( N  e.  Fin  /\  R  e.  V ) 
 ->  ( 0g `  G )  =  ( 0g `  A ) )
 
Theoremmatinvg 19520 The matrix ring has the same additive inverse as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.)
 |-  A  =  ( N Mat 
 R )   &    |-  G  =  ( R freeLMod  ( N  X.  N ) )   =>    |-  ( ( N  e.  Fin  /\  R  e.  V ) 
 ->  ( invg `  G )  =  ( invg `  A ) )
 
Theoremmat0op 19521* Value of a zero matrix as operation. (Contributed by AV, 2-Dec-2018.)
 |-  A  =  ( N Mat 
 R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  ( 0g `  A )  =  ( i  e.  N ,  j  e.  N  |->  .0.  ) )
 
Theoremmatsca2 19522 The scalars of the matrix ring are the underlying ring. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  A  =  ( N Mat 
 R )   =>    |-  ( ( N  e.  Fin  /\  R  e.  V ) 
 ->  R  =  (Scalar `  A ) )
 
Theoremmatbas2 19523 The base set of the matrix ring as a set exponential. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 16-Dec-2018.)
 |-  A  =  ( N Mat 
 R )   &    |-  K  =  (
 Base `  R )   =>    |-  ( ( N  e.  Fin  /\  R  e.  V )  ->  ( K 
 ^m  ( N  X.  N ) )  =  ( Base `  A )
 )
 
Theoremmatbas2i 19524 A matrix is a function. (Contributed by Stefan O'Rear, 11-Sep-2015.)
 |-  A  =  ( N Mat 
 R )   &    |-  K  =  (
 Base `  R )   &    |-  B  =  ( Base `  A )   =>    |-  ( M  e.  B  ->  M  e.  ( K  ^m  ( N  X.  N ) ) )
 
Theoremmatbas2d 19525* The base set of the matrix ring as a mapping operation. (Contributed by Stefan O'Rear, 11-Jul-2018.)
 |-  A  =  ( N Mat 
 R )   &    |-  K  =  (
 Base `  R )   &    |-  B  =  ( Base `  A )   &    |-  ( ph  ->  N  e.  Fin )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ( ph  /\  x  e.  N  /\  y  e.  N )  ->  C  e.  K )   =>    |-  ( ph  ->  ( x  e.  N ,  y  e.  N  |->  C )  e.  B )
 
Theoremeqmat 19526* Two square matrices of the same dimension are equal if they have the same entries. (Contributed by AV, 25-Sep-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   =>    |-  ( ( X  e.  B  /\  Y  e.  B )  ->  ( X  =  Y  <->  A. i  e.  N  A. j  e.  N  ( i X j )  =  ( i Y j ) ) )
 
Theoremmatecl 19527 Each entry (according to Wikipedia "Matrix (mathematics)", 30-Dec-2018, https://en.wikipedia.org/wiki/Matrix_(mathematics)#Definition (or element or component or coefficient or cell) of a matrix is an element of the underlying ring. (Contributed by AV, 16-Dec-2018.)
 |-  A  =  ( N Mat 
 R )   &    |-  K  =  (
 Base `  R )   =>    |-  ( ( I  e.  N  /\  J  e.  N  /\  M  e.  ( Base `  A )
 )  ->  ( I M J )  e.  K )
 
Theoremmatecld 19528 Each entry (according to Wikipedia "Matrix (mathematics)", 30-Dec-2018, https://en.wikipedia.org/wiki/Matrix_(mathematics)#Definition (or element or component or coefficient or cell) of a matrix is an element of the underlying ring, deduction form. (Contributed by AV, 27-Nov-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  K  =  (
 Base `  R )   &    |-  B  =  ( Base `  A )   &    |-  ( ph  ->  I  e.  N )   &    |-  ( ph  ->  J  e.  N )   &    |-  ( ph  ->  M  e.  B )   =>    |-  ( ph  ->  ( I M J )  e.  K )
 
Theoremmatplusg2 19529 Addition in the matrix ring is cell-wise. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  .+b  =  ( +g  `  A )   &    |-  .+  =  ( +g  `  R )   =>    |-  (
 ( X  e.  B  /\  Y  e.  B ) 
 ->  ( X  .+b  Y )  =  ( X  oF  .+  Y ) )
 
Theoremmatvsca2 19530 Scalar multiplication in the matrix ring is cell-wise. (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  K  =  ( Base `  R )   &    |-  .x.  =  ( .s `  A )   &    |-  .X. 
 =  ( .r `  R )   &    |-  C  =  ( N  X.  N )   =>    |-  ( ( X  e.  K  /\  Y  e.  B )  ->  ( X  .x.  Y )  =  ( ( C  X.  { X } )  oF  .X.  Y ) )
 
Theoremmatlmod 19531 The matrix ring is a linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.)
 |-  A  =  ( N Mat 
 R )   =>    |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  A  e.  LMod )
 
Theoremmatgrp 19532 The matrix ring is a group. (Contributed by AV, 21-Dec-2019.)
 |-  A  =  ( N Mat 
 R )   =>    |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  A  e.  Grp )
 
Theoremmatvscl 19533 Closure of the scalar multiplication in the matrix ring. (lmodvscl 18186 analog.) (Contributed by AV, 27-Nov-2019.)
 |-  K  =  ( Base `  R )   &    |-  A  =  ( N Mat  R )   &    |-  B  =  ( Base `  A )   &    |-  .x.  =  ( .s `  A )   =>    |-  ( ( ( N  e.  Fin  /\  R  e.  Ring
 )  /\  ( C  e.  K  /\  X  e.  B ) )  ->  ( C  .x.  X )  e.  B )
 
Theoremmatsubg 19534 The matrix ring has the same addition as its underlying group. (Contributed by AV, 2-Aug-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  G  =  ( R freeLMod  ( N  X.  N ) )   =>    |-  ( ( N  e.  Fin  /\  R  e.  V ) 
 ->  ( -g `  G )  =  ( -g `  A ) )
 
Theoremmatplusgcell 19535 Addition in the matrix ring is cell-wise. (Contributed by AV, 2-Aug-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  .+b  =  ( +g  `  A )   &    |-  .+  =  ( +g  `  R )   =>    |-  (
 ( ( X  e.  B  /\  Y  e.  B )  /\  ( I  e.  N  /\  J  e.  N ) )  ->  ( I ( X  .+b  Y ) J )  =  ( ( I X J )  .+  ( I Y J ) ) )
 
Theoremmatsubgcell 19536 Subtraction in the matrix ring is cell-wise. (Contributed by AV, 2-Aug-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  S  =  ( -g `  A )   &    |-  .-  =  ( -g `  R )   =>    |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  B ) 
 /\  ( I  e.  N  /\  J  e.  N ) )  ->  ( I ( X S Y ) J )  =  ( ( I X J )  .-  ( I Y J ) ) )
 
Theoremmatinvgcell 19537 Additive inversion in the matrix ring is cell-wise. (Contributed by AV, 17-Nov-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  V  =  ( invg `  R )   &    |-  W  =  ( invg `  A )   =>    |-  ( ( R  e.  Ring  /\  X  e.  B  /\  ( I  e.  N  /\  J  e.  N ) )  ->  ( I
 ( W `  X ) J )  =  ( V `  ( I X J ) ) )
 
Theoremmatvscacell 19538 Scalar multiplication in the matrix ring is cell-wise. (Contributed by AV, 7-Aug-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  K  =  ( Base `  R )   &    |-  .x.  =  ( .s `  A )   &    |-  .X. 
 =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  ( X  e.  K  /\  Y  e.  B ) 
 /\  ( I  e.  N  /\  J  e.  N ) )  ->  ( I ( X  .x.  Y ) J )  =  ( X  .X.  ( I Y J ) ) )
 
Theoremmatgsum 19539* Finite commutative sums in a matrix algebra are taken componentwise. (Contributed by AV, 26-Sep-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  .0.  =  ( 0g `  A )   &    |-  ( ph  ->  N  e.  Fin )   &    |-  ( ph  ->  J  e.  W )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ( ph  /\  y  e.  J )  ->  (
 i  e.  N ,  j  e.  N  |->  U )  e.  B )   &    |-  ( ph  ->  ( y  e.  J  |->  ( i  e.  N ,  j  e.  N  |->  U ) ) finSupp  .0.  )   =>    |-  ( ph  ->  ( A  gsumg  ( y  e.  J  |->  ( i  e.  N ,  j  e.  N  |->  U ) ) )  =  ( i  e.  N ,  j  e.  N  |->  ( R  gsumg  ( y  e.  J  |->  U ) ) ) )
 
11.2.3  The matrix algebra

The main result of this subsection are the theorems showing that  ( N Mat  R ) is a ring (see matring 19545) and an associative algebra (see matassa 19546). Additionally, theorems for the identity matrix and transposed matrices are provided.

 
Theoremmatmulr 19540 Multiplication in the matrix algebra. (Contributed by Stefan O'Rear, 4-Sep-2015.)
 |-  A  =  ( N Mat 
 R )   &    |-  .x.  =  ( R maMul  <. N ,  N ,  N >. )   =>    |-  ( ( N  e.  Fin  /\  R  e.  V ) 
 ->  .x.  =  ( .r
 `  A ) )
 
Theoremmamumat1cl 19541* The identity matrix (as operation in maps-to notation) is a matrix. (Contributed by Stefan O'Rear, 2-Sep-2015.)
 |-  B  =  ( Base `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  .1.  =  ( 1r `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  I  =  ( i  e.  M ,  j  e.  M  |->  if (
 i  =  j ,  .1.  ,  .0.  )
 )   &    |-  ( ph  ->  M  e.  Fin )   =>    |-  ( ph  ->  I  e.  ( B  ^m  ( M  X.  M ) ) )
 
Theoremmat1comp 19542* The components of the identity matrix (as operation in maps-to notation). (Contributed by AV, 22-Jul-2019.)
 |-  B  =  ( Base `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  .1.  =  ( 1r `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  I  =  ( i  e.  M ,  j  e.  M  |->  if (
 i  =  j ,  .1.  ,  .0.  )
 )   &    |-  ( ph  ->  M  e.  Fin )   =>    |-  ( ( A  e.  M  /\  J  e.  M )  ->  ( A I J )  =  if ( A  =  J ,  .1.  ,  .0.  )
 )
 
Theoremmamulid 19543* The identity matrix (as operation in maps-to notation) is a left identity (for any matrix with the same number of rows). (Contributed by Stefan O'Rear, 3-Sep-2015.) (Proof shortened by AV, 22-Jul-2019.)
 |-  B  =  ( Base `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  .1.  =  ( 1r `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  I  =  ( i  e.  M ,  j  e.  M  |->  if (
 i  =  j ,  .1.  ,  .0.  )
 )   &    |-  ( ph  ->  M  e.  Fin )   &    |-  ( ph  ->  N  e.  Fin )   &    |-  F  =  ( R maMul  <. M ,  M ,  N >. )   &    |-  ( ph  ->  X  e.  ( B  ^m  ( M  X.  N ) ) )   =>    |-  ( ph  ->  ( I F X )  =  X )
 
Theoremmamurid 19544* The identity matrix (as operation in maps-to notation) is a right identity (for any matrix with the same number of columns). (Contributed by Stefan O'Rear, 3-Sep-2015.) (Proof shortened by AV, 22-Jul-2019.)
 |-  B  =  ( Base `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  .1.  =  ( 1r `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  I  =  ( i  e.  M ,  j  e.  M  |->  if (
 i  =  j ,  .1.  ,  .0.  )
 )   &    |-  ( ph  ->  M  e.  Fin )   &    |-  ( ph  ->  N  e.  Fin )   &    |-  F  =  ( R maMul  <. N ,  M ,  M >. )   &    |-  ( ph  ->  X  e.  ( B  ^m  ( N  X.  M ) ) )   =>    |-  ( ph  ->  ( X F I )  =  X )
 
Theoremmatring 19545 Existence of the matrix ring, see also the statement in [Lang] p. 504: "For a given integer n > 0 the set of square n x n matrices form a ring." (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  A  =  ( N Mat 
 R )   =>    |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  A  e.  Ring )
 
Theoremmatassa 19546 Existence of the matrix algebra, see also the statement in [Lang] p. 505:"Then Matn(R) is an algebra over R" . (Contributed by Stefan O'Rear, 5-Sep-2015.)
 |-  A  =  ( N Mat 
 R )   =>    |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  A  e. AssAlg )
 
Theoremmatmulcell 19547* Multiplication in the matrix ring for a single cell of a matrix. (Contributed by AV, 17-Nov-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  .x.  =  ( .r `  R )   &    |-  .X. 
 =  ( .r `  A )   =>    |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  B ) 
 /\  ( I  e.  N  /\  J  e.  N ) )  ->  ( I ( X  .X.  Y ) J )  =  ( R  gsumg  ( j  e.  N  |->  ( ( I X j ) ( .r
 `  R ) ( j Y J ) ) ) ) )
 
Theoremmpt2matmul 19548* Multiplication of two N x N matrices given in maps-to notation. (Contributed by AV, 29-Oct-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  R )   &    |-  .X.  =  ( .r `  A )   &    |-  .x. 
 =  ( .r `  R )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  N  e.  Fin )   &    |-  X  =  ( i  e.  N ,  j  e.  N  |->  C )   &    |-  Y  =  ( i  e.  N ,  j  e.  N  |->  E )   &    |-  ( ( ph  /\  i  e.  N  /\  j  e.  N )  ->  C  e.  B )   &    |-  ( ( ph  /\  i  e.  N  /\  j  e.  N )  ->  E  e.  B )   &    |-  ( ( ph  /\  ( k  =  i 
 /\  m  =  j ) )  ->  D  =  C )   &    |-  ( ( ph  /\  ( m  =  i 
 /\  l  =  j ) )  ->  F  =  E )   &    |-  ( ( ph  /\  k  e.  N  /\  m  e.  N )  ->  D  e.  U )   &    |-  ( ( ph  /\  m  e.  N  /\  l  e.  N )  ->  F  e.  W )   =>    |-  ( ph  ->  ( X  .X.  Y )  =  ( k  e.  N ,  l  e.  N  |->  ( R  gsumg  ( m  e.  N  |->  ( D  .x.  F ) ) ) ) )
 
Theoremmat1 19549* Value of an identity matrix, see also the statement in [Lang] p. 504: "The unit element of the ring of n x n matrices is the matrix In ... whose components are equal to 0 except on the diagonal, in which case they are equal to 1.". (Contributed by Stefan O'Rear, 7-Sep-2015.)
 |-  A  =  ( N Mat 
 R )   &    |-  .1.  =  ( 1r `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  ( 1r `  A )  =  ( i  e.  N ,  j  e.  N  |->  if ( i  =  j ,  .1.  ,  .0.  ) ) )
 
Theoremmat1ov 19550 Entries of an identity matrix, deduction form. (Contributed by Stefan O'Rear, 10-Jul-2018.)
 |-  A  =  ( N Mat 
 R )   &    |-  .1.  =  ( 1r `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  ( ph  ->  N  e.  Fin )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  I  e.  N )   &    |-  ( ph  ->  J  e.  N )   &    |-  U  =  ( 1r `  A )   =>    |-  ( ph  ->  ( I U J )  =  if ( I  =  J ,  .1.  ,  .0.  ) )
 
Theoremmat1bas 19551 The identity matrix is a matrix. (Contributed by AV, 15-Feb-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  .1.  =  ( 1r `  ( N Mat  R ) )   =>    |-  ( ( R  e.  Ring  /\  N  e.  Fin )  ->  .1.  e.  B )
 
Theoremmatsc 19552* The identity matrix multiplied with a scalar. (Contributed by Stefan O'Rear, 16-Jul-2018.)
 |-  A  =  ( N Mat 
 R )   &    |-  K  =  (
 Base `  R )   &    |-  .x.  =  ( .s `  A )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  L  e.  K )  ->  ( L  .x.  ( 1r `  A ) )  =  ( i  e.  N ,  j  e.  N  |->  if ( i  =  j ,  L ,  .0.  ) ) )
 
Theoremofco2 19553 Distribution law for the function operation and the composition of functions. (Contributed by Stefan O'Rear, 17-Jul-2018.)
 |-  ( ( ( F  e.  _V  /\  G  e.  _V )  /\  ( Fun  H  /\  ( F  o.  H )  e. 
 _V  /\  ( G  o.  H )  e.  _V ) )  ->  ( ( F  oF R G )  o.  H )  =  ( ( F  o.  H )  oF R ( G  o.  H ) ) )
 
Theoremoftpos 19554 The transposition of the value of a function operation for two functions is the value of the function operation for the two functions transposed. (Contributed by Stefan O'Rear, 17-Jul-2018.)
 |-  ( ( F  e.  V  /\  G  e.  W )  -> tpos  ( F  oF R G )  =  (tpos 
 F  oF Rtpos 
 G ) )
 
Theoremmattposcl 19555 The transpose of a square matrix is a square matrix of the same size. (Contributed by SO, 9-Jul-2018.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   =>    |-  ( M  e.  B  -> tpos  M  e.  B )
 
Theoremmattpostpos 19556 The transpose of the transpose of a square matrix is the square matrix itself. (Contributed by SO, 17-Jul-2018.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   =>    |-  ( M  e.  B  -> tpos tpos  M  =  M )
 
Theoremmattposvs 19557 The transposition of a matrix multiplied with a scalar equals the transposed matrix multiplied with the scalar, see also the statement in [Lang] p. 505. (Contributed by Stefan O'Rear, 17-Jul-2018.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  K  =  ( Base `  R )   &    |-  .x.  =  ( .s `  A )   =>    |-  ( ( X  e.  K  /\  Y  e.  B )  -> tpos  ( X  .x.  Y )  =  ( X  .x. tpos  Y ) )
 
Theoremmattpos1 19558 The transposition of the identity matrix is the identity matrix. (Contributed by Stefan O'Rear, 17-Jul-2018.)
 |-  A  =  ( N Mat 
 R )   &    |-  .1.  =  ( 1r `  A )   =>    |-  ( ( N  e.  Fin  /\  R  e.  Ring )  -> tpos 
 .1.  =  .1.  )
 
Theoremtposmap 19559 The transposition of an I X J -matrix is a J X I -matrix, see also the statement in [Lang] p. 505. (Contributed by Stefan O'Rear, 9-Jul-2018.)
 |-  ( A  e.  ( B  ^m  ( I  X.  J ) )  -> tpos  A  e.  ( B  ^m  ( J  X.  I ) ) )
 
Theoremmamutpos 19560 Behavior of transposes in matrix products, see also the statement in [Lang] p. 505. (Contributed by Stefan O'Rear, 9-Jul-2018.)
 |-  F  =  ( R maMul  <. M ,  N ,  P >. )   &    |-  G  =  ( R maMul  <. P ,  N ,  M >. )   &    |-  B  =  (
 Base `  R )   &    |-  ( ph  ->  R  e.  CRing )   &    |-  ( ph  ->  M  e.  Fin )   &    |-  ( ph  ->  N  e.  Fin )   &    |-  ( ph  ->  P  e.  Fin )   &    |-  ( ph  ->  X  e.  ( B  ^m  ( M  X.  N ) ) )   &    |-  ( ph  ->  Y  e.  ( B  ^m  ( N  X.  P ) ) )   =>    |-  ( ph  -> tpos  ( X F Y )  =  (tpos  Y Gtpos  X ) )
 
Theoremmattposm 19561 Multiplying two transposed matrices results in the transposition of the product of the two matrices. (Contributed by Stefan O'Rear, 17-Jul-2018.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  .x.  =  ( .r `  A )   =>    |-  ( ( R  e.  CRing  /\  X  e.  B  /\  Y  e.  B )  -> tpos  ( X  .x.  Y )  =  (tpos  Y  .x. tpos  X ) )
 
Theoremmatgsumcl 19562* Closure of a group sum over the diagonal coefficients of a square matrix over a commutative ring. (Contributed by AV, 29-Dec-2018.) (Proof shortened by AV, 23-Jul-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  U  =  (mulGrp `  R )   =>    |-  (
 ( R  e.  CRing  /\  M  e.  B ) 
 ->  ( U  gsumg  ( r  e.  N  |->  ( r M r ) ) )  e.  ( Base `  R )
 )
 
Theoremmadetsumid 19563* The identity summand in the Leibniz' formula of a determinant for a square matrix over a commutative ring. (Contributed by AV, 29-Dec-2018.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  U  =  (mulGrp `  R )   &    |-  Y  =  ( ZRHom `  R )   &    |-  S  =  (pmSgn `  N )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( R  e.  CRing  /\  M  e.  B  /\  P  =  (  _I  |`  N )
 )  ->  ( (
 ( Y  o.  S ) `  P )  .x.  ( U  gsumg  ( r  e.  N  |->  ( ( P `  r ) M r ) ) ) )  =  ( U  gsumg  ( r  e.  N  |->  ( r M r ) ) ) )
 
Theoremmatepmcl 19564* Each entry of a matrix with an index as permutation of the other is an element of the underlying ring. (Contributed by AV, 1-Jan-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  P  =  ( Base `  ( SymGrp `  N ) )   =>    |-  ( ( R  e.  Ring  /\  Q  e.  P  /\  M  e.  B )  ->  A. n  e.  N  ( ( Q `  n ) M n )  e.  ( Base `  R ) )
 
Theoremmatepm2cl 19565* Each entry of a matrix with an index as permutation of the other is an element of the underlying ring. (Contributed by AV, 1-Jan-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  P  =  ( Base `  ( SymGrp `  N ) )   =>    |-  ( ( R  e.  Ring  /\  Q  e.  P  /\  M  e.  B )  ->  A. n  e.  N  ( n M ( Q `
  n ) )  e.  ( Base `  R ) )
 
Theoremmadetsmelbas 19566* A summand of the determinant of a matrix belongs to the underlying ring. (Contributed by AV, 1-Jan-2019.)
 |-  P  =  ( Base `  ( SymGrp `  N )
 )   &    |-  S  =  (pmSgn `  N )   &    |-  Y  =  ( ZRHom `  R )   &    |-  A  =  ( N Mat  R )   &    |-  B  =  ( Base `  A )   &    |-  G  =  (mulGrp `  R )   =>    |-  ( ( R  e.  CRing  /\  M  e.  B  /\  Q  e.  P )  ->  ( ( ( Y  o.  S ) `  Q ) ( .r
 `  R ) ( G  gsumg  ( n  e.  N  |->  ( ( Q `  n ) M n ) ) ) )  e.  ( Base `  R ) )
 
Theoremmadetsmelbas2 19567* A summand of the determinant of a matrix belongs to the underlying ring. (Contributed by AV, 1-Jan-2019.)
 |-  P  =  ( Base `  ( SymGrp `  N )
 )   &    |-  S  =  (pmSgn `  N )   &    |-  Y  =  ( ZRHom `  R )   &    |-  A  =  ( N Mat  R )   &    |-  B  =  ( Base `  A )   &    |-  G  =  (mulGrp `  R )   =>    |-  ( ( R  e.  CRing  /\  M  e.  B  /\  Q  e.  P )  ->  ( ( ( Y  o.  S ) `  Q ) ( .r
 `  R ) ( G  gsumg  ( n  e.  N  |->  ( n M ( Q `
  n ) ) ) ) )  e.  ( Base `  R )
 )
 
11.2.4  Matrices of dimension 0 and 1

As already mentioned before, and shown in mat0dimbas0 19568, the empty set is the sole zero-dimensional matrix (also called "empty matrix", see Wikipedia https://en.wikipedia.org/wiki/Matrix_(mathematics)#Empty_matrices). In the following, some properties of the empty matrix are shown, especially that the empty matrix over an arbitrary ring forms a commutative ring, see mat0dimcrng 19572.

For the one-dimensional case, it can be shown that a ring of matrices with dimension 1 is isomorphic to the underlying ring, see mat1ric 19589.

 
Theoremmat0dimbas0 19568 The empty set is the one and only matrix of dimension 0, called "the empty matrix". (Contributed by AV, 27-Feb-2019.)
 |-  ( R  e.  V  ->  ( Base `  ( (/) Mat  R ) )  =  { (/) } )
 
Theoremmat0dim0 19569 The zero of the algebra of matrices with dimension 0. (Contributed by AV, 6-Aug-2019.)
 |-  A  =  ( (/) Mat  R )   =>    |-  ( R  e.  Ring  ->  ( 0g `  A )  =  (/) )
 
Theoremmat0dimid 19570 The identity of the algebra of matrices with dimension 0. (Contributed by AV, 6-Aug-2019.)
 |-  A  =  ( (/) Mat  R )   =>    |-  ( R  e.  Ring  ->  ( 1r `  A )  =  (/) )
 
Theoremmat0dimscm 19571 The scalar multiplication in the algebra of matrices with dimension 0. (Contributed by AV, 6-Aug-2019.)
 |-  A  =  ( (/) Mat  R )   =>    |-  ( ( R  e.  Ring  /\  X  e.  ( Base `  R ) )  ->  ( X ( .s `  A ) (/) )  =  (/) )
 
Theoremmat0dimcrng 19572 The algebra of matrices with dimension 0 (over an arbitrary ring!) is a commutative ring. (Contributed by AV, 10-Aug-2019.)
 |-  A  =  ( (/) Mat  R )   =>    |-  ( R  e.  Ring  ->  A  e.  CRing )
 
Theoremmat1dimelbas 19573* A matrix with dimension 1 is an ordered pair with an ordered pair (of the one and only pair of indices) as first component. (Contributed by AV, 15-Aug-2019.)
 |-  A  =  ( { E } Mat  R )   &    |-  B  =  ( Base `  R )   &    |-  O  =  <. E ,  E >.   =>    |-  ( ( R  e.  Ring  /\  E  e.  V ) 
 ->  ( M  e.  ( Base `  A )  <->  E. r  e.  B  M  =  { <. O ,  r >. } ) )
 
Theoremmat1dimbas 19574 A matrix with dimension 1 is an ordered pair with an ordered pair (of the one and only pair of indices) as first component. (Contributed by AV, 15-Aug-2019.)
 |-  A  =  ( { E } Mat  R )   &    |-  B  =  ( Base `  R )   &    |-  O  =  <. E ,  E >.   =>    |-  ( ( R  e.  Ring  /\  E  e.  V  /\  X  e.  B )  ->  { <. O ,  X >. }  e.  ( Base `  A ) )
 
Theoremmat1dim0 19575 The zero of the algebra of matrices with dimension 1. (Contributed by AV, 15-Aug-2019.)
 |-  A  =  ( { E } Mat  R )   &    |-  B  =  ( Base `  R )   &    |-  O  =  <. E ,  E >.   =>    |-  ( ( R  e.  Ring  /\  E  e.  V ) 
 ->  ( 0g `  A )  =  { <. O ,  ( 0g `  R )
 >. } )
 
Theoremmat1dimid 19576 The identity of the algebra of matrices with dimension 1. (Contributed by AV, 15-Aug-2019.)
 |-  A  =  ( { E } Mat  R )   &    |-  B  =  ( Base `  R )   &    |-  O  =  <. E ,  E >.   =>    |-  ( ( R  e.  Ring  /\  E  e.  V ) 
 ->  ( 1r `  A )  =  { <. O ,  ( 1r `  R )
 >. } )
 
Theoremmat1dimscm 19577 The scalar multiplication in the algebra of matrices with dimension 1. (Contributed by AV, 16-Aug-2019.)
 |-  A  =  ( { E } Mat  R )   &    |-  B  =  ( Base `  R )   &    |-  O  =  <. E ,  E >.   =>    |-  ( ( ( R  e.  Ring  /\  E  e.  V )  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  ( X ( .s `  A ) { <. O ,  Y >. } )  =  { <. O ,  ( X ( .r `  R ) Y )
 >. } )
 
Theoremmat1dimmul 19578 The ring multiplication in the algebra of matrices with dimension 1. (Contributed by AV, 16-Aug-2019.)
 |-  A  =  ( { E } Mat  R )   &    |-  B  =  ( Base `  R )   &    |-  O  =  <. E ,  E >.   =>    |-  ( ( ( R  e.  Ring  /\  E  e.  V )  /\  ( X  e.  B  /\  Y  e.  B ) )  ->  ( { <. O ,  X >. }  ( .r `  A ) { <. O ,  Y >. } )  =  { <. O ,  ( X ( .r `  R ) Y )
 >. } )
 
Theoremmat1dimcrng 19579 The algebra of matrices with dimension 1 over a commutative ring is a commutative ring. (Contributed by AV, 16-Aug-2019.)
 |-  A  =  ( { E } Mat  R )   &    |-  B  =  ( Base `  R )   &    |-  O  =  <. E ,  E >.   =>    |-  ( ( R  e.  CRing  /\  E  e.  V ) 
 ->  A  e.  CRing )
 
Theoremmat1f1o 19580* There is a 1-1 function from a ring onto the ring of matrices with dimension 1 over this ring. (Contributed by AV, 22-Dec-2019.)
 |-  K  =  ( Base `  R )   &    |-  A  =  ( { E } Mat  R )   &    |-  B  =  ( Base `  A )   &    |-  O  =  <. E ,  E >.   &    |-  F  =  ( x  e.  K  |->  { <. O ,  x >. } )   =>    |-  (
 ( R  e.  Ring  /\  E  e.  V ) 
 ->  F : K -1-1-onto-> B )
 
Theoremmat1rhmval 19581* The value of the ring homomorphism 
F. (Contributed by AV, 22-Dec-2019.)
 |-  K  =  ( Base `  R )   &    |-  A  =  ( { E } Mat  R )   &    |-  B  =  ( Base `  A )   &    |-  O  =  <. E ,  E >.   &    |-  F  =  ( x  e.  K  |->  { <. O ,  x >. } )   =>    |-  (
 ( R  e.  Ring  /\  E  e.  V  /\  X  e.  K )  ->  ( F `  X )  =  { <. O ,  X >. } )
 
Theoremmat1rhmelval 19582* The value of the ring homomorphism 
F. (Contributed by AV, 22-Dec-2019.)
 |-  K  =  ( Base `  R )   &    |-  A  =  ( { E } Mat  R )   &    |-  B  =  ( Base `  A )   &    |-  O  =  <. E ,  E >.   &    |-  F  =  ( x  e.  K  |->  { <. O ,  x >. } )   =>    |-  (
 ( R  e.  Ring  /\  E  e.  V  /\  X  e.  K )  ->  ( E ( F `
  X ) E )  =  X )
 
Theoremmat1rhmcl 19583* The value of the ring homomorphism 
F is a matrix with dimension 1. (Contributed by AV, 22-Dec-2019.)
 |-  K  =  ( Base `  R )   &    |-  A  =  ( { E } Mat  R )   &    |-  B  =  ( Base `  A )   &    |-  O  =  <. E ,  E >.   &    |-  F  =  ( x  e.  K  |->  { <. O ,  x >. } )   =>    |-  (
 ( R  e.  Ring  /\  E  e.  V  /\  X  e.  K )  ->  ( F `  X )  e.  B )
 
Theoremmat1f 19584* There is a function from a ring to the ring of matrices with dimension 1 over this ring. (Contributed by AV, 22-Dec-2019.)
 |-  K  =  ( Base `  R )   &    |-  A  =  ( { E } Mat  R )   &    |-  B  =  ( Base `  A )   &    |-  O  =  <. E ,  E >.   &    |-  F  =  ( x  e.  K  |->  { <. O ,  x >. } )   =>    |-  (
 ( R  e.  Ring  /\  E  e.  V ) 
 ->  F : K --> B )
 
Theoremmat1ghm 19585* There is a group homomorphism from the additive group of a ring to the additive group of the ring of matrices with dimension 1 over this ring. (Contributed by AV, 22-Dec-2019.)
 |-  K  =  ( Base `  R )   &    |-  A  =  ( { E } Mat  R )   &    |-  B  =  ( Base `  A )   &    |-  O  =  <. E ,  E >.   &    |-  F  =  ( x  e.  K  |->  { <. O ,  x >. } )   =>    |-  (
 ( R  e.  Ring  /\  E  e.  V ) 
 ->  F  e.  ( R 
 GrpHom  A ) )
 
Theoremmat1mhm 19586* There is a monoid homomorphism from the multiplicative group of a ring to the multiplicative group of the ring of matrices with dimension 1 over this ring. (Contributed by AV, 22-Dec-2019.)
 |-  K  =  ( Base `  R )   &    |-  A  =  ( { E } Mat  R )   &    |-  B  =  ( Base `  A )   &    |-  O  =  <. E ,  E >.   &    |-  F  =  ( x  e.  K  |->  { <. O ,  x >. } )   &    |-  M  =  (mulGrp `  R )   &    |-  N  =  (mulGrp `  A )   =>    |-  (
 ( R  e.  Ring  /\  E  e.  V ) 
 ->  F  e.  ( M MndHom  N ) )
 
Theoremmat1rhm 19587* There is a ring homomorphism from a ring to the ring of matrices with dimension 1 over this ring. (Contributed by AV, 22-Dec-2019.)
 |-  K  =  ( Base `  R )   &    |-  A  =  ( { E } Mat  R )   &    |-  B  =  ( Base `  A )   &    |-  O  =  <. E ,  E >.   &    |-  F  =  ( x  e.  K  |->  { <. O ,  x >. } )   =>    |-  (
 ( R  e.  Ring  /\  E  e.  V ) 
 ->  F  e.  ( R RingHom  A ) )
 
Theoremmat1rngiso 19588* There is a ring isomorphism from a ring to the ring of matrices with dimension 1 over this ring. (Contributed by AV, 22-Dec-2019.)
 |-  K  =  ( Base `  R )   &    |-  A  =  ( { E } Mat  R )   &    |-  B  =  ( Base `  A )   &    |-  O  =  <. E ,  E >.   &    |-  F  =  ( x  e.  K  |->  { <. O ,  x >. } )   =>    |-  (
 ( R  e.  Ring  /\  E  e.  V ) 
 ->  F  e.  ( R RingIso  A ) )
 
Theoremmat1ric 19589 A ring is isomorphic to the ring of matrices with dimension 1 over this ring. (Contributed by AV, 30-Dec-2019.)
 |-  A  =  ( { E } Mat  R )   =>    |-  (
 ( R  e.  Ring  /\  E  e.  V ) 
 ->  R  ~=r  A )
 
11.2.5  The subalgebras of diagonal and scalar matrices

According to Wikipedia ("Diagonal Matrix", 8-Dec-2019, https://en.wikipedia.org/wiki/Diagonal_matrix): "In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices." The diagonal matrices are mentioned in [Lang] p. 576, but without giving them a dedicated definition. Furthermore, "A diagonal matrix with all its main diagonal entries equal is a scalar matrix, that is, a scalar multiple  la  .*  I of the identity matrix  I. Its effect on a vector is scalar multiplication by  la [see scmatscm 19615!]". The scalar multiples of the identity matrix are mentioned in [Lang] p. 504, but without giving them a special name.

The main results of this subsection are the definitions of the sets of diagonal and scalar matrices (df-dmat 19592 and df-scmat 19593), basic properties of (elements of) these sets, and theorems showing that the diagonal matrices are a subring of the ring of square matrices (dmatsrng 19603), that the scalar matrices are a subring of the ring of square matrices (scmatsrng 19622), that the scalar matrices are a subring of the ring of diagonal matrices (scmatsrng1 19625) and that the ring of scalar matrices (over a commutative ring) is a commutative ring (scmatcrng 19623).

 
Syntaxcdmat 19590 Extend class notation for the algebra of diagonal matrices.
 class DMat
 
Syntaxcscmat 19591 Extend class notation for the algebra of scalar matrices.
 class ScMat
 
Definitiondf-dmat 19592* Define the set of n x n diagonal (square) matrices over a set (usually a ring) r, see definition in [Roman] p. 4 or Definition 3.12 in [Hefferon] p. 240. (Contributed by AV, 8-Dec-2019.)
 |- DMat  =  ( n  e.  Fin ,  r  e.  _V  |->  { m  e.  ( Base `  ( n Mat  r ) )  |  A. i  e.  n  A. j  e.  n  ( i  =/=  j  ->  ( i m j )  =  ( 0g `  r
 ) ) } )
 
Definitiondf-scmat 19593* Define the algebra of n x n scalar matrices over a set (usually a ring) r, see definition in [Connell] p. 57: "A scalar matrix is a diagonal matrix for which all the diagonal terms are equal, i.e., a matrix of the form cIn";. (Contributed by AV, 8-Dec-2019.)
 |- ScMat  =  ( n  e.  Fin ,  r  e.  _V  |->  [_ ( n Mat  r )  /  a ]_ { m  e.  ( Base `  a )  |  E. c  e.  ( Base `  r ) m  =  ( c ( .s `  a ) ( 1r `  a
 ) ) } )
 
Theoremdmatval 19594* The set of  N x  N diagonal matrices over (a ring)  R. (Contributed by AV, 8-Dec-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  .0.  =  ( 0g `  R )   &    |-  D  =  ( N DMat 
 R )   =>    |-  ( ( N  e.  Fin  /\  R  e.  V ) 
 ->  D  =  { m  e.  B  |  A. i  e.  N  A. j  e.  N  ( i  =/=  j  ->  ( i m j )  =  .0.  ) } )
 
Theoremdmatel 19595* A  N x  N diagonal matrix over (a ring)  R. (Contributed by AV, 18-Dec-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  .0.  =  ( 0g `  R )   &    |-  D  =  ( N DMat 
 R )   =>    |-  ( ( N  e.  Fin  /\  R  e.  V ) 
 ->  ( M  e.  D  <->  ( M  e.  B  /\  A. i  e.  N  A. j  e.  N  (
 i  =/=  j  ->  ( i M j )  =  .0.  ) ) ) )
 
Theoremdmatmat 19596 An  N x  N diagonal matrix over (the ring)  R is an  N x  N matrix over (the ring)  R. (Contributed by AV, 18-Dec-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  .0.  =  ( 0g `  R )   &    |-  D  =  ( N DMat 
 R )   =>    |-  ( ( N  e.  Fin  /\  R  e.  V ) 
 ->  ( M  e.  D  ->  M  e.  B ) )
 
Theoremdmatid 19597 The identity matrix is a diagonal matrix. (Contributed by AV, 19-Aug-2019.) (Revised by AV, 18-Dec-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  .0.  =  ( 0g `  R )   &    |-  D  =  ( N DMat 
 R )   =>    |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  ( 1r `  A )  e.  D )
 
Theoremdmatelnd 19598 An extradiagonal entry of a diagonal matrix is equal to zero. (Contributed by AV, 19-Aug-2019.) (Revised by AV, 18-Dec-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  .0.  =  ( 0g `  R )   &    |-  D  =  ( N DMat 
 R )   =>    |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  X  e.  D ) 
 /\  ( I  e.  N  /\  J  e.  N  /\  I  =/=  J ) )  ->  ( I X J )  =  .0.  )
 
Theoremdmatmul 19599* The product of two diagonal matrices. (Contributed by AV, 19-Aug-2019.) (Revised by AV, 18-Dec-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  .0.  =  ( 0g `  R )   &    |-  D  =  ( N DMat 
 R )   =>    |-  ( ( ( N  e.  Fin  /\  R  e.  Ring
 )  /\  ( X  e.  D  /\  Y  e.  D ) )  ->  ( X ( .r `  A ) Y )  =  ( x  e.  N ,  y  e.  N  |->  if ( x  =  y ,  ( ( x X y ) ( .r `  R ) ( x Y y ) ) ,  .0.  ) ) )
 
Theoremdmatsubcl 19600 The difference of two diagonal matrices is a diagonal matrix. (Contributed by AV, 19-Aug-2019.) (Revised by AV, 18-Dec-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  .0.  =  ( 0g `  R )   &    |-  D  =  ( N DMat 
 R )   =>    |-  ( ( ( N  e.  Fin  /\  R  e.  Ring
 )  /\  ( X  e.  D  /\  Y  e.  D ) )  ->  ( X ( -g `  A ) Y )  e.  D )
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