HomeHome Metamath Proof Explorer
Theorem List (p. 196 of 402)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-26569)
  Hilbert Space Explorer  Hilbert Space Explorer
(26570-28092)
  Users' Mathboxes  Users' Mathboxes
(28093-40161)
 

Theorem List for Metamath Proof Explorer - 19501-19600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmat1f1o 19501* 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 19502* 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 19503* 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 19504* 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 19505* 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 19506* 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 19507* 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 19508* 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 19509* 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 19510 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 19536!]". 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 19513 and df-scmat 19514), basic properties of (elements of) these sets, and theorems showing that the diagonal matrices are a subring of the ring of square matrices (dmatsrng 19524), that the scalar matrices are a subring of the ring of square matrices (scmatsrng 19543), that the scalar matrices are a subring of the ring of diagonal matrices (scmatsrng1 19546) and that the ring of scalar matrices (over a commutative ring) is a commutative ring (scmatcrng 19544).

 
Syntaxcdmat 19511 Extend class notation for the algebra of diagonal matrices.
 class DMat
 
Syntaxcscmat 19512 Extend class notation for the algebra of scalar matrices.
 class ScMat
 
Definitiondf-dmat 19513* 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 19514* 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 19515* 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 19516* 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 19517 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 19518 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 19519 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 19520* 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 19521 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 )
 
Theoremdmatsgrp 19522 The set of diagonal matrices is a subgroup of the matrix group/algebra. (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 )   =>    |-  ( ( R  e.  Ring  /\  N  e.  Fin )  ->  D  e.  (SubGrp `  A ) )
 
Theoremdmatmulcl 19523 The product of two diagonal matrices is a diagonal matrix. (Contributed by AV, 20-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 )  e.  D )
 
Theoremdmatsrng 19524 The set of diagonal matrices is a subring of the matrix ring/algebra. (Contributed by AV, 20-Aug-2019.) (Revised by AV, 18-Dec-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  .0.  =  ( 0g `  R )   &    |-  D  =  ( N DMat 
 R )   =>    |-  ( ( R  e.  Ring  /\  N  e.  Fin )  ->  D  e.  (SubRing `  A ) )
 
Theoremdmatcrng 19525 The subring of diagonal matrices (over a commutative ring) is a commutative ring . (Contributed by AV, 20-Aug-2019.) (Revised by AV, 18-Dec-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  .0.  =  ( 0g `  R )   &    |-  D  =  ( N DMat 
 R )   &    |-  C  =  ( As  D )   =>    |-  ( ( R  e.  CRing  /\  N  e.  Fin )  ->  C  e.  CRing )
 
Theoremdmatscmcl 19526 The multiplication of a diagonal matrix with a scalar is a diagonal matrix. (Contributed by AV, 19-Dec-2019.)
 |-  K  =  ( Base `  R )   &    |-  A  =  ( N Mat  R )   &    |-  B  =  ( Base `  A )   &    |-  .*  =  ( .s `  A )   &    |-  D  =  ( N DMat 
 R )   =>    |-  ( ( ( N  e.  Fin  /\  R  e.  Ring
 )  /\  ( C  e.  K  /\  M  e.  D ) )  ->  ( C  .*  M )  e.  D )
 
Theoremscmatval 19527* The set of  N x  N scalar matrices over (a ring)  R. (Contributed by AV, 18-Dec-2019.)
 |-  K  =  ( Base `  R )   &    |-  A  =  ( N Mat  R )   &    |-  B  =  ( Base `  A )   &    |-  .1.  =  ( 1r `  A )   &    |- 
 .x.  =  ( .s `  A )   &    |-  S  =  ( N ScMat  R )   =>    |-  ( ( N  e.  Fin  /\  R  e.  V )  ->  S  =  { m  e.  B  |  E. c  e.  K  m  =  ( c  .x.  .1.  ) } )
 
Theoremscmatel 19528* An  N x  N scalar matrix over (a ring)  R. (Contributed by AV, 18-Dec-2019.)
 |-  K  =  ( Base `  R )   &    |-  A  =  ( N Mat  R )   &    |-  B  =  ( Base `  A )   &    |-  .1.  =  ( 1r `  A )   &    |- 
 .x.  =  ( .s `  A )   &    |-  S  =  ( N ScMat  R )   =>    |-  ( ( N  e.  Fin  /\  R  e.  V )  ->  ( M  e.  S  <->  ( M  e.  B  /\  E. c  e.  K  M  =  ( c  .x.  .1.  )
 ) ) )
 
Theoremscmatscmid 19529* A scalar matrix can be expressed as a multiplication of a scalar with the identity matrix. (Contributed by AV, 30-Oct-2019.) (Revised by AV, 18-Dec-2019.)
 |-  K  =  ( Base `  R )   &    |-  A  =  ( N Mat  R )   &    |-  B  =  ( Base `  A )   &    |-  .1.  =  ( 1r `  A )   &    |- 
 .x.  =  ( .s `  A )   &    |-  S  =  ( N ScMat  R )   =>    |-  ( ( N  e.  Fin  /\  R  e.  V  /\  M  e.  S )  ->  E. c  e.  K  M  =  ( c  .x.  .1.  ) )
 
Theoremscmatscmide 19530 An entry of a scalar matrix expressed as a multiplication of a scalar with the identity matrix. (Contributed by AV, 30-Oct-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |- 
 .1.  =  ( 1r `  A )   &    |-  .*  =  ( .s `  A )   =>    |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  C  e.  B ) 
 /\  ( I  e.  N  /\  J  e.  N ) )  ->  ( I ( C  .*  .1.  ) J )  =  if ( I  =  J ,  C ,  .0.  ) )
 
Theoremscmatscmiddistr 19531 Distributive law for scalar and ring multiplication for scalar matrices expressed as multiplications of a scalar with the identity matrix. (Contributed by AV, 19-Dec-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |- 
 .1.  =  ( 1r `  A )   &    |-  .*  =  ( .s `  A )   &    |-  .x. 
 =  ( .r `  R )   &    |-  .X.  =  ( .r `  A )   =>    |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  ( S  e.  B  /\  T  e.  B ) )  ->  ( ( S  .*  .1.  )  .X.  ( T  .*  .1.  )
 )  =  ( ( S  .x.  T )  .*  .1.  ) )
 
Theoremscmatmat 19532 An  N x  N scalar 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 )   &    |-  S  =  ( N ScMat  R )   =>    |-  ( ( N  e.  Fin  /\  R  e.  V ) 
 ->  ( M  e.  S  ->  M  e.  B ) )
 
Theoremscmate 19533* An entry of an  N x  N scalar matrix over the ring  R. (Contributed by AV, 18-Dec-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  S  =  ( N ScMat  R )   &    |-  K  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  S ) 
 /\  ( I  e.  N  /\  J  e.  N ) )  ->  E. c  e.  K  ( I M J )  =  if ( I  =  J ,  c ,  .0.  ) )
 
Theoremscmatmats 19534* The set of an  N x  N scalar matrices over the ring  R expressed as a subset of  N x  N matrices over the ring  R with certain properties for their entries. (Contributed by AV, 31-Oct-2019.) (Revised by AV, 19-Dec-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  S  =  ( N ScMat  R )   &    |-  K  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  S  =  { m  e.  B  |  E. c  e.  K  A. i  e.  N  A. j  e.  N  ( i m j )  =  if ( i  =  j ,  c ,  .0.  ) } )
 
TheoremscmateALT 19535* Alternate proof of scmate 19533: An entry of an  N x  N scalar matrix over the ring  R. This prove makes use of scmatmats 19534 but is longer and requires more distinct variables. (Contributed by AV, 19-Dec-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  S  =  ( N ScMat  R )   &    |-  K  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( ( N  e.  Fin  /\  R  e.  Ring  /\  M  e.  S ) 
 /\  ( I  e.  N  /\  J  e.  N ) )  ->  E. c  e.  K  ( I M J )  =  if ( I  =  J ,  c ,  .0.  ) )
 
Theoremscmatscm 19536* The multiplication of a matrix with a scalar matrix corresponds to a scalar multiplication. (Contributed by AV, 28-Dec-2019.)
 |-  K  =  ( Base `  R )   &    |-  A  =  ( N Mat  R )   &    |-  B  =  ( Base `  A )   &    |-  .*  =  ( .s `  A )   &    |- 
 .X.  =  ( .r `  A )   &    |-  S  =  ( N ScMat  R )   =>    |-  ( ( ( N  e.  Fin  /\  R  e.  Ring )  /\  C  e.  S )  ->  E. c  e.  K  A. m  e.  B  ( C  .X.  m )  =  ( c  .*  m ) )
 
Theoremscmatid 19537 The identity matrix is a scalar matrix. (Contributed by AV, 20-Aug-2019.) (Revised by AV, 18-Dec-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  E  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  S  =  ( N ScMat  R )   =>    |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  ( 1r `  A )  e.  S )
 
Theoremscmatdmat 19538 A scalar matrix is a diagonal matrix. (Contributed by AV, 20-Aug-2019.) (Revised by AV, 19-Dec-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  E  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  S  =  ( N ScMat  R )   &    |-  D  =  ( N DMat  R )   =>    |-  ( ( N  e.  Fin  /\  R  e.  Ring
 )  ->  ( M  e.  S  ->  M  e.  D ) )
 
Theoremscmataddcl 19539 The sum of two scalar matrices is a scalar matrix. (Contributed by AV, 25-Dec-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  E  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  S  =  ( N ScMat  R )   =>    |-  ( ( ( N  e.  Fin  /\  R  e.  Ring
 )  /\  ( X  e.  S  /\  Y  e.  S ) )  ->  ( X ( +g  `  A ) Y )  e.  S )
 
Theoremscmatsubcl 19540 The difference of two scalar matrices is a scalar matrix. (Contributed by AV, 20-Aug-2019.) (Revised by AV, 19-Dec-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  E  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  S  =  ( N ScMat  R )   =>    |-  ( ( ( N  e.  Fin  /\  R  e.  Ring
 )  /\  ( X  e.  S  /\  Y  e.  S ) )  ->  ( X ( -g `  A ) Y )  e.  S )
 
Theoremscmatmulcl 19541 The product of two scalar matrices is a scalar matrix. (Contributed by AV, 21-Aug-2019.) (Revised by AV, 19-Dec-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  E  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  S  =  ( N ScMat  R )   =>    |-  ( ( ( N  e.  Fin  /\  R  e.  Ring
 )  /\  ( X  e.  S  /\  Y  e.  S ) )  ->  ( X ( .r `  A ) Y )  e.  S )
 
Theoremscmatsgrp 19542 The set of scalar matrices is a subgroup of the matrix group/algebra. (Contributed by AV, 20-Aug-2019.) (Revised by AV, 19-Dec-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  E  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  S  =  ( N ScMat  R )   =>    |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  S  e.  (SubGrp `  A ) )
 
Theoremscmatsrng 19543 The set of scalar matrices is a subring of the matrix ring/algebra. (Contributed by AV, 21-Aug-2019.) (Revised by AV, 19-Dec-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  E  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  S  =  ( N ScMat  R )   =>    |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  S  e.  (SubRing `  A ) )
 
Theoremscmatcrng 19544 The subring of scalar matrices (over a commutative ring) is a commutative ring. (Contributed by AV, 21-Aug-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  E  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  S  =  ( N ScMat  R )   &    |-  C  =  ( As  S )   =>    |-  ( ( N  e.  Fin  /\  R  e.  CRing )  ->  C  e.  CRing )
 
Theoremscmatsgrp1 19545 The set of scalar matrices is a subgroup of the group/ring of diagonal matrices. (Contributed by AV, 21-Aug-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  E  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  S  =  ( N ScMat  R )   &    |-  D  =  ( N DMat  R )   &    |-  C  =  ( As  D )   =>    |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  S  e.  (SubGrp `  C ) )
 
Theoremscmatsrng1 19546 The set of scalar matrices is a subring of the ring of diagonal matrices. (Contributed by AV, 21-Aug-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  E  =  ( Base `  R )   &    |-  .0.  =  ( 0g `  R )   &    |-  S  =  ( N ScMat  R )   &    |-  D  =  ( N DMat  R )   &    |-  C  =  ( As  D )   =>    |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  S  e.  (SubRing `  C ) )
 
Theoremsmatvscl 19547 Closure of the scalar multiplication in the ring of scalar matrices. (matvscl 19454 analog.) (Contributed by AV, 24-Dec-2019.)
 |-  K  =  ( Base `  R )   &    |-  A  =  ( N Mat  R )   &    |-  S  =  ( N ScMat  R )   &    |-  .*  =  ( .s `  A )   =>    |-  ( ( ( N  e.  Fin  /\  R  e.  Ring
 )  /\  ( C  e.  K  /\  X  e.  S ) )  ->  ( C  .*  X )  e.  S )
 
Theoremscmatlss 19548 The set of scalar matrices is a linear subspace of the matrix algebra. (Contributed by AV, 25-Dec-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  S  =  ( N ScMat  R )   =>    |-  ( ( N  e.  Fin  /\  R  e.  Ring
 )  ->  S  e.  ( LSubSp `  A )
 )
 
Theoremscmatstrbas 19549 The set of scalar matrices is the base set of the ring of corresponding scalar matrices. (Contributed by AV, 26-Dec-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  C  =  ( N ScMat  R )   &    |-  S  =  ( As  C )   =>    |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  ( Base `  S )  =  C )
 
Theoremscmatrhmval 19550* The value of the ring homomorphism 
F. (Contributed by AV, 22-Dec-2019.)
 |-  K  =  ( Base `  R )   &    |-  A  =  ( N Mat  R )   &    |-  .1.  =  ( 1r `  A )   &    |- 
 .*  =  ( .s
 `  A )   &    |-  F  =  ( x  e.  K  |->  ( x  .*  .1.  )
 )   =>    |-  ( ( R  e.  V  /\  X  e.  K )  ->  ( F `  X )  =  ( X  .*  .1.  ) )
 
Theoremscmatrhmcl 19551* The value of the ring homomorphism 
F is a scalar matrix. (Contributed by AV, 22-Dec-2019.)
 |-  K  =  ( Base `  R )   &    |-  A  =  ( N Mat  R )   &    |-  .1.  =  ( 1r `  A )   &    |- 
 .*  =  ( .s
 `  A )   &    |-  F  =  ( x  e.  K  |->  ( x  .*  .1.  )
 )   &    |-  C  =  ( N ScMat  R )   =>    |-  ( ( N  e.  Fin  /\  R  e.  Ring  /\  X  e.  K )  ->  ( F `  X )  e.  C )
 
Theoremscmatf 19552* There is a function from a ring to any ring of scalar matrices over this ring. (Contributed by AV, 25-Dec-2019.)
 |-  K  =  ( Base `  R )   &    |-  A  =  ( N Mat  R )   &    |-  .1.  =  ( 1r `  A )   &    |- 
 .*  =  ( .s
 `  A )   &    |-  F  =  ( x  e.  K  |->  ( x  .*  .1.  )
 )   &    |-  C  =  ( N ScMat  R )   =>    |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  F : K --> C )
 
Theoremscmatfo 19553* There is a function from a ring onto any ring of scalar matrices over this ring. (Contributed by AV, 26-Dec-2019.)
 |-  K  =  ( Base `  R )   &    |-  A  =  ( N Mat  R )   &    |-  .1.  =  ( 1r `  A )   &    |- 
 .*  =  ( .s
 `  A )   &    |-  F  =  ( x  e.  K  |->  ( x  .*  .1.  )
 )   &    |-  C  =  ( N ScMat  R )   =>    |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  F : K -onto-> C )
 
Theoremscmatf1 19554* There is a 1-1 function from a ring to any ring of scalar matrices with positive dimension over this ring. (Contributed by AV, 25-Dec-2019.)
 |-  K  =  ( Base `  R )   &    |-  A  =  ( N Mat  R )   &    |-  .1.  =  ( 1r `  A )   &    |- 
 .*  =  ( .s
 `  A )   &    |-  F  =  ( x  e.  K  |->  ( x  .*  .1.  )
 )   &    |-  C  =  ( N ScMat  R )   =>    |-  ( ( N  e.  Fin  /\  N  =/=  (/)  /\  R  e.  Ring )  ->  F : K -1-1-> C )
 
Theoremscmatf1o 19555* There is a bijection between a ring and any ring of scalar matrices with positive dimension over this ring. (Contributed by AV, 26-Dec-2019.)
 |-  K  =  ( Base `  R )   &    |-  A  =  ( N Mat  R )   &    |-  .1.  =  ( 1r `  A )   &    |- 
 .*  =  ( .s
 `  A )   &    |-  F  =  ( x  e.  K  |->  ( x  .*  .1.  )
 )   &    |-  C  =  ( N ScMat  R )   =>    |-  ( ( N  e.  Fin  /\  N  =/=  (/)  /\  R  e.  Ring )  ->  F : K -1-1-onto-> C )
 
Theoremscmatghm 19556* There is a group homomorphism from the additive group of a ring to the additive group of the ring of scalar matrices over this ring. (Contributed by AV, 22-Dec-2019.)
 |-  K  =  ( Base `  R )   &    |-  A  =  ( N Mat  R )   &    |-  .1.  =  ( 1r `  A )   &    |- 
 .*  =  ( .s
 `  A )   &    |-  F  =  ( x  e.  K  |->  ( x  .*  .1.  )
 )   &    |-  C  =  ( N ScMat  R )   &    |-  S  =  ( As  C )   =>    |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  F  e.  ( R 
 GrpHom  S ) )
 
Theoremscmatmhm 19557* There is a monoid homomorphism from the multiplicative group of a ring to the multiplicative group of the ring of scalar matrices over this ring. (Contributed by AV, 29-Dec-2019.)
 |-  K  =  ( Base `  R )   &    |-  A  =  ( N Mat  R )   &    |-  .1.  =  ( 1r `  A )   &    |- 
 .*  =  ( .s
 `  A )   &    |-  F  =  ( x  e.  K  |->  ( x  .*  .1.  )
 )   &    |-  C  =  ( N ScMat  R )   &    |-  S  =  ( As  C )   &    |-  M  =  (mulGrp `  R )   &    |-  T  =  (mulGrp `  S )   =>    |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  F  e.  ( M MndHom  T ) )
 
Theoremscmatrhm 19558* There is a ring homomorphism from a ring to the ring of scalar matrices over this ring. (Contributed by AV, 29-Dec-2019.)
 |-  K  =  ( Base `  R )   &    |-  A  =  ( N Mat  R )   &    |-  .1.  =  ( 1r `  A )   &    |- 
 .*  =  ( .s
 `  A )   &    |-  F  =  ( x  e.  K  |->  ( x  .*  .1.  )
 )   &    |-  C  =  ( N ScMat  R )   &    |-  S  =  ( As  C )   =>    |-  ( ( N  e.  Fin  /\  R  e.  Ring )  ->  F  e.  ( R RingHom  S ) )
 
Theoremscmatrngiso 19559* There is a ring isomorphism from a ring to the ring of scalar matrices over this ring with positive dimension. (Contributed by AV, 29-Dec-2019.)
 |-  K  =  ( Base `  R )   &    |-  A  =  ( N Mat  R )   &    |-  .1.  =  ( 1r `  A )   &    |- 
 .*  =  ( .s
 `  A )   &    |-  F  =  ( x  e.  K  |->  ( x  .*  .1.  )
 )   &    |-  C  =  ( N ScMat  R )   &    |-  S  =  ( As  C )   =>    |-  ( ( N  e.  Fin  /\  N  =/=  (/)  /\  R  e.  Ring )  ->  F  e.  ( R RingIso  S )
 )
 
Theoremscmatric 19560 A ring is isomorphic to every ring of scalar matrices over this ring with positive dimension. (Contributed by AV, 29-Dec-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  C  =  ( N ScMat  R )   &    |-  S  =  ( As  C )   =>    |-  ( ( N  e.  Fin  /\  N  =/=  (/)  /\  R  e.  Ring )  ->  R  ~=r 
 S )
 
Theoremmat0scmat 19561 The empty matrix over a ring is a scalar matrix (and therefore, by scmatdmat 19538, also a diagonal matrix). (Contributed by AV, 21-Dec-2019.)
 |-  ( R  e.  Ring  ->  (/) 
 e.  ( (/) ScMat  R ) )
 
Theoremmat1scmat 19562 A 1-dimensional matrix over a ring is always a scalar matrix (and therefore, by scmatdmat 19538, also a diagonal matrix). (Contributed by AV, 21-Dec-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   =>    |-  ( ( N  e.  V  /\  ( # `
  N )  =  1  /\  R  e.  Ring
 )  ->  ( M  e.  B  ->  M  e.  ( N ScMat  R ) ) )
 
11.2.6  Multiplication of a matrix with a "column vector"

The module of  n-dimensional "column vectors" over a ring  r is the  n-dimensional free module over a ring  r, which is the product of  n -many copies of the ring with componentwise addition and multiplication. Although a "column vector" could also be defined as n x 1 -matrix (according to Wikipedia "Row and column vectors", 22-Feb-2019, https://en.wikipedia.org/wiki/Row_and_column_vectors: "In linear algebra, a column vector [... ] is an m x 1 matrix, that is, a matrix consisting of a single column of m elements"), which would allow for using the matrix multiplication df-mamu 19407 for multiplying a matrix with a column vector, it seems more natural to use the definition of a free (left) module, avoiding to provide a singleton as  1-dimensional index set for the column, and to introduce a new operator df-mvmul 19564 for the multiplication of a matrix with a column vector. In most cases, it is sufficient to regard members of  ( ( Base `  R )  ^m  N ) as "column vectors", because  ( ( Base `  R )  ^m  N ) is the base set of  ( R freeLMod  N ), see frlmbasmap 19320. See also the statements in [Lang] p. 508.

 
Syntaxcmvmul 19563 Syntax for the operator for the multiplication of a vector with a matrix.
 class maVecMul
 
Definitiondf-mvmul 19564* The operator which multiplies an M x N -matrix with an N-dimensional vector. (Contributed by AV, 23-Feb-2019.)
 |- maVecMul  =  ( r  e.  _V ,  o  e.  _V  |->  [_ ( 1st `  o
 )  /  m ]_ [_ ( 2nd `  o )  /  n ]_ ( x  e.  ( ( Base `  r
 )  ^m  ( m  X.  n ) ) ,  y  e.  ( (
 Base `  r )  ^m  n )  |->  ( i  e.  m  |->  ( r 
 gsumg  ( j  e.  n  |->  ( ( i x j ) ( .r
 `  r ) ( y `  j ) ) ) ) ) ) )
 
Theoremmvmulfval 19565* Functional value of the matrix vector multiplication operator. (Contributed by AV, 23-Feb-2019.)
 |- 
 .X.  =  ( R maVecMul  <. M ,  N >. )   &    |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  M  e.  Fin )   &    |-  ( ph  ->  N  e.  Fin )   =>    |-  ( ph  ->  .X. 
 =  ( x  e.  ( B  ^m  ( M  X.  N ) ) ,  y  e.  ( B  ^m  N )  |->  ( i  e.  M  |->  ( R  gsumg  ( j  e.  N  |->  ( ( i x j )  .x.  (
 y `  j )
 ) ) ) ) ) )
 
Theoremmvmulval 19566* Multiplication of a vector with a matrix. (Contributed by AV, 23-Feb-2019.)
 |- 
 .X.  =  ( R maVecMul  <. M ,  N >. )   &    |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  M  e.  Fin )   &    |-  ( ph  ->  N  e.  Fin )   &    |-  ( ph  ->  X  e.  ( B  ^m  ( M  X.  N ) ) )   &    |-  ( ph  ->  Y  e.  ( B  ^m  N ) )   =>    |-  ( ph  ->  ( X  .X.  Y )  =  ( i  e.  M  |->  ( R  gsumg  ( j  e.  N  |->  ( ( i X j )  .x.  ( Y `  j ) ) ) ) ) )
 
Theoremmvmulfv 19567* A cell/element in the vector resulting from a multiplication of a vector with a matrix. (Contributed by AV, 23-Feb-2019.)
 |- 
 .X.  =  ( R maVecMul  <. M ,  N >. )   &    |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  M  e.  Fin )   &    |-  ( ph  ->  N  e.  Fin )   &    |-  ( ph  ->  X  e.  ( B  ^m  ( M  X.  N ) ) )   &    |-  ( ph  ->  Y  e.  ( B  ^m  N ) )   &    |-  ( ph  ->  I  e.  M )   =>    |-  ( ph  ->  ( ( X  .X.  Y ) `  I )  =  ( R  gsumg  ( j  e.  N  |->  ( ( I X j )  .x.  ( Y `  j ) ) ) ) )
 
Theoremmavmulval 19568* Multiplication of a vector with a square matrix. (Contributed by AV, 23-Feb-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  .X.  =  ( R maVecMul 
 <. N ,  N >. )   &    |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  N  e.  Fin )   &    |-  ( ph  ->  X  e.  ( Base `  A ) )   &    |-  ( ph  ->  Y  e.  ( B  ^m  N ) )   =>    |-  ( ph  ->  ( X  .X.  Y )  =  ( i  e.  N  |->  ( R  gsumg  ( j  e.  N  |->  ( ( i X j )  .x.  ( Y `  j ) ) ) ) ) )
 
Theoremmavmulfv 19569* A cell/element in the vector resulting from a multiplication of a vector with a square matrix. (Contributed by AV, 6-Dec-2018.) (Revised by AV, 18-Feb-2019.) (Revised by AV, 23-Feb-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  .X.  =  ( R maVecMul 
 <. N ,  N >. )   &    |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  ( ph  ->  R  e.  V )   &    |-  ( ph  ->  N  e.  Fin )   &    |-  ( ph  ->  X  e.  ( Base `  A ) )   &    |-  ( ph  ->  Y  e.  ( B  ^m  N ) )   &    |-  ( ph  ->  I  e.  N )   =>    |-  ( ph  ->  (
 ( X  .X.  Y ) `  I )  =  ( R  gsumg  ( j  e.  N  |->  ( ( I X j )  .x.  ( Y `  j ) ) ) ) )
 
Theoremmavmulcl 19570 Multiplication of an NxN matrix with an N-dimensional vector results in an N-dimensional vector. (Contributed by AV, 6-Dec-2018.) (Revised by AV, 23-Feb-2019.) (Proof shortened by AV, 23-Jul-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  .X.  =  ( R maVecMul 
 <. N ,  N >. )   &    |-  B  =  ( Base `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  N  e.  Fin )   &    |-  ( ph  ->  X  e.  ( Base `  A ) )   &    |-  ( ph  ->  Y  e.  ( B  ^m  N ) )   =>    |-  ( ph  ->  ( X  .X.  Y )  e.  ( B  ^m  N ) )
 
Theorem1mavmul 19571 Multiplication of the identity NxN matrix with an N-dimensional vector results in the vector itself. (Contributed by AV, 9-Feb-2019.) (Revised by AV, 23-Feb-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  R )   &    |-  .x.  =  ( R maVecMul  <. N ,  N >. )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  N  e.  Fin )   &    |-  ( ph  ->  Y  e.  ( B  ^m  N ) )   =>    |-  ( ph  ->  ( ( 1r `  A )  .x.  Y )  =  Y )
 
Theoremmavmulass 19572 Associativity of the multiplication of two NxN matrices with an N-dimensional vector. (Contributed by AV, 9-Feb-2019.) (Revised by AV, 25-Feb-2019.) (Proof shortened by AV, 22-Jul-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  R )   &    |-  .x.  =  ( R maVecMul  <. N ,  N >. )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  N  e.  Fin )   &    |-  ( ph  ->  Y  e.  ( B  ^m  N ) )   &    |-  .X. 
 =  ( R maMul  <. N ,  N ,  N >. )   &    |-  ( ph  ->  X  e.  ( Base `  A )
 )   &    |-  ( ph  ->  Z  e.  ( Base `  A )
 )   =>    |-  ( ph  ->  (
 ( X  .X.  Z )  .x.  Y )  =  ( X  .x.  ( Z  .x.  Y ) ) )
 
Theoremmavmuldm 19573 The domain of the matrix vector multiplication function. (Contributed by AV, 27-Feb-2019.)
 |-  B  =  ( Base `  R )   &    |-  C  =  ( B  ^m  ( M  X.  N ) )   &    |-  D  =  ( B  ^m  N )   &    |-  .x.  =  ( R maVecMul 
 <. M ,  N >. )   =>    |-  ( ( R  e.  V  /\  M  e.  Fin  /\  N  e.  Fin )  ->  dom  .x.  =  ( C  X.  D ) )
 
Theoremmavmulsolcl 19574 Every solution of the equation  A * X  =  Y for a matrix  A and a vector  B is a vector. (Contributed by AV, 27-Feb-2019.)
 |-  B  =  ( Base `  R )   &    |-  C  =  ( B  ^m  ( M  X.  N ) )   &    |-  D  =  ( B  ^m  N )   &    |-  .x.  =  ( R maVecMul 
 <. M ,  N >. )   &    |-  E  =  ( B  ^m  M )   =>    |-  ( ( ( M  e.  Fin  /\  N  e.  Fin  /\  M  =/=  (/) )  /\  ( R  e.  V  /\  Y  e.  E ) )  ->  ( ( A  .x.  X )  =  Y  ->  X  e.  D ) )
 
Theoremmavmul0 19575 Multiplication of a 0-dimensional matrix with a 0-dimensional vector. (Contributed by AV, 28-Feb-2019.)
 |- 
 .x.  =  ( R maVecMul  <. N ,  N >. )   =>    |-  ( ( N  =  (/)  /\  R  e.  V ) 
 ->  ( (/)  .x.  (/) )  =  (/) )
 
Theoremmavmul0g 19576 The result of the 0-dimensional multiplication of a matrix with a vector is always the empty set. (Contributed by AV, 1-Mar-2019.)
 |- 
 .x.  =  ( R maVecMul  <. N ,  N >. )   =>    |-  ( ( N  =  (/)  /\  R  e.  V ) 
 ->  ( X  .x.  Y )  =  (/) )
 
Theoremmvmumamul1 19577* The multiplication of an MxN matrix with an N-dimensional vector corresponds to the matrix multiplication of an MxN matrix with an Nx1 matrix. (Contributed by AV, 14-Mar-2019.)
 |- 
 .X.  =  ( R maMul  <. M ,  N ,  { (/) } >. )   &    |-  .x.  =  ( R maVecMul  <. M ,  N >. )   &    |-  B  =  (
 Base `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  M  e.  Fin )   &    |-  ( ph  ->  N  e.  Fin )   &    |-  ( ph  ->  A  e.  ( B  ^m  ( M  X.  N ) ) )   &    |-  ( ph  ->  Y  e.  ( B  ^m  N ) )   &    |-  ( ph  ->  Z  e.  ( B  ^m  ( N  X.  { (/) } )
 ) )   =>    |-  ( ph  ->  ( A. j  e.  N  ( Y `  j )  =  ( j Z (/) )  ->  A. i  e.  M  ( ( A 
 .x.  Y ) `  i
 )  =  ( i ( A  .X.  Z ) (/) ) ) )
 
Theoremmavmumamul1 19578* The multiplication of an NxN matrix with an N-dimensional vector corresponds to the matrix multiplication of an NxN matrix with an Nx1 matrix. (Contributed by AV, 14-Mar-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  .X.  =  ( R maMul  <. N ,  N ,  { (/) } >. )   &    |-  .x.  =  ( R maVecMul  <. N ,  N >. )   &    |-  B  =  (
 Base `  R )   &    |-  ( ph  ->  R  e.  Ring )   &    |-  ( ph  ->  N  e.  Fin )   &    |-  ( ph  ->  X  e.  ( Base `  A ) )   &    |-  ( ph  ->  Y  e.  ( B  ^m  N ) )   &    |-  ( ph  ->  Z  e.  ( B  ^m  ( N  X.  { (/) } ) ) )   =>    |-  ( ph  ->  ( A. j  e.  N  ( Y `  j )  =  ( j Z (/) )  ->  A. i  e.  N  ( ( X  .x.  Y ) `  i )  =  ( i ( X  .X.  Z ) (/) ) ) )
 
11.2.7  Replacement functions for a square matrix
 
Syntaxcmarrep 19579 Syntax for the row replacing function for a square matrix.
 class matRRep
 
SyntaxcmatrepV 19580 Syntax for the function replacing a column of a matrix by a vector.
 class matRepV
 
Definitiondf-marrep 19581* Define the matrices whose k-th row is replaced by 0's and an arbitrary element of the underlying ring at the l-th column. (Contributed by AV, 12-Feb-2019.)
 |- matRRep  =  ( n  e.  _V ,  r  e.  _V  |->  ( m  e.  ( Base `  ( n Mat  r
 ) ) ,  s  e.  ( Base `  r )  |->  ( k  e.  n ,  l  e.  n  |->  ( i  e.  n ,  j  e.  n  |->  if ( i  =  k ,  if ( j  =  l ,  s ,  ( 0g `  r
 ) ) ,  (
 i m j ) ) ) ) ) )
 
Definitiondf-marepv 19582* Function replacing a column of a matrix by a vector. (Contributed by AV, 9-Feb-2019.) (Revised by AV, 26-Feb-2019.)
 |- matRepV  =  ( n  e.  _V ,  r  e.  _V  |->  ( m  e.  ( Base `  ( n Mat  r
 ) ) ,  v  e.  ( ( Base `  r
 )  ^m  n )  |->  ( k  e.  n  |->  ( i  e.  n ,  j  e.  n  |->  if ( j  =  k ,  ( v `  i ) ,  (
 i m j ) ) ) ) ) )
 
Theoremmarrepfval 19583* First substitution for the definition of the matrix row replacement function. (Contributed by AV, 12-Feb-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  Q  =  ( N matRRep  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  Q  =  ( m  e.  B ,  s  e.  ( Base `  R )  |->  ( k  e.  N ,  l  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( i  =  k ,  if ( j  =  l ,  s ,  .0.  ) ,  (
 i m j ) ) ) ) )
 
Theoremmarrepval0 19584* Second substitution for the definition of the matrix row replacement function. (Contributed by AV, 12-Feb-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  Q  =  ( N matRRep  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( M  e.  B  /\  S  e.  ( Base `  R ) ) 
 ->  ( M Q S )  =  ( k  e.  N ,  l  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( i  =  k ,  if (
 j  =  l ,  S ,  .0.  ) ,  ( i M j ) ) ) ) )
 
Theoremmarrepval 19585* Third substitution for the definition of the matrix row replacement function. (Contributed by AV, 12-Feb-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  Q  =  ( N matRRep  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( ( M  e.  B  /\  S  e.  ( Base `  R )
 )  /\  ( K  e.  N  /\  L  e.  N ) )  ->  ( K ( M Q S ) L )  =  ( i  e.  N ,  j  e.  N  |->  if ( i  =  K ,  if (
 j  =  L ,  S ,  .0.  ) ,  ( i M j ) ) ) )
 
Theoremmarrepeval 19586 An entry of a matrix with a replaced row. (Contributed by AV, 12-Feb-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  Q  =  ( N matRRep  R )   &    |-  .0.  =  ( 0g `  R )   =>    |-  ( ( ( M  e.  B  /\  S  e.  ( Base `  R )
 )  /\  ( K  e.  N  /\  L  e.  N )  /\  ( I  e.  N  /\  J  e.  N ) )  ->  ( I ( K ( M Q S ) L ) J )  =  if ( I  =  K ,  if ( J  =  L ,  S ,  .0.  ) ,  ( I M J ) ) )
 
Theoremmarrepcl 19587 Closure of the row replacement function for square matrices. (Contributed by AV, 13-Feb-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   =>    |-  ( ( ( R  e.  Ring  /\  M  e.  B  /\  S  e.  ( Base `  R )
 )  /\  ( K  e.  N  /\  L  e.  N ) )  ->  ( K ( M ( N matRRep  R ) S ) L )  e.  B )
 
Theoremmarepvfval 19588* First substitution for the definition of the function replacing a column of a matrix by a vector. (Contributed by AV, 14-Feb-2019.) (Revised by AV, 26-Feb-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  Q  =  ( N matRepV  R )   &    |-  V  =  ( ( Base `  R )  ^m  N )   =>    |-  Q  =  ( m  e.  B ,  v  e.  V  |->  ( k  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( j  =  k ,  ( v `
  i ) ,  ( i m j ) ) ) ) )
 
Theoremmarepvval0 19589* Second substitution for the definition of the function replacing a column of a matrix by a vector. (Contributed by AV, 14-Feb-2019.) (Revised by AV, 26-Feb-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  Q  =  ( N matRepV  R )   &    |-  V  =  ( ( Base `  R )  ^m  N )   =>    |-  ( ( M  e.  B  /\  C  e.  V )  ->  ( M Q C )  =  ( k  e.  N  |->  ( i  e.  N ,  j  e.  N  |->  if ( j  =  k ,  ( C `  i ) ,  (
 i M j ) ) ) ) )
 
Theoremmarepvval 19590* Third substitution for the definition of the function replacing a column of a matrix by a vector. (Contributed by AV, 14-Feb-2019.) (Revised by AV, 26-Feb-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  Q  =  ( N matRepV  R )   &    |-  V  =  ( ( Base `  R )  ^m  N )   =>    |-  ( ( M  e.  B  /\  C  e.  V  /\  K  e.  N )  ->  ( ( M Q C ) `
  K )  =  ( i  e.  N ,  j  e.  N  |->  if ( j  =  K ,  ( C `  i
 ) ,  ( i M j ) ) ) )
 
Theoremmarepveval 19591 An entry of a matrix with a replaced column. (Contributed by AV, 14-Feb-2019.) (Revised by AV, 26-Feb-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  Q  =  ( N matRepV  R )   &    |-  V  =  ( ( Base `  R )  ^m  N )   =>    |-  ( ( ( M  e.  B  /\  C  e.  V  /\  K  e.  N )  /\  ( I  e.  N  /\  J  e.  N ) )  ->  ( I
 ( ( M Q C ) `  K ) J )  =  if ( J  =  K ,  ( C `  I
 ) ,  ( I M J ) ) )
 
Theoremmarepvcl 19592 Closure of the column replacement function for square matrices. (Contributed by AV, 14-Feb-2019.) (Revised by AV, 26-Feb-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  V  =  ( ( Base `  R )  ^m  N )   =>    |-  ( ( R  e.  Ring  /\  ( M  e.  B  /\  C  e.  V  /\  K  e.  N ) )  ->  ( ( M ( N matRepV  R ) C ) `
  K )  e.  B )
 
Theoremma1repvcl 19593 Closure of the column replacement function for identity matrices. (Contributed by AV, 15-Feb-2019.) (Revised by AV, 26-Feb-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  V  =  ( ( Base `  R )  ^m  N )   &    |-  .1.  =  ( 1r `  A )   =>    |-  ( ( ( R  e.  Ring  /\  N  e.  Fin )  /\  ( C  e.  V  /\  K  e.  N ) )  ->  ( (  .1.  ( N matRepV  R ) C ) `
  K )  e.  B )
 
Theoremma1repveval 19594 An entry of an identity matrix with a replaced column. (Contributed by AV, 16-Feb-2019.) (Revised by AV, 26-Feb-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  V  =  ( ( Base `  R )  ^m  N )   &    |-  .1.  =  ( 1r `  A )   &    |- 
 .0.  =  ( 0g `  R )   &    |-  E  =  ( (  .1.  ( N matRepV  R ) C ) `
  K )   =>    |-  ( ( R  e.  Ring  /\  ( M  e.  B  /\  C  e.  V  /\  K  e.  N )  /\  ( I  e.  N  /\  J  e.  N ) )  ->  ( I E J )  =  if ( J  =  K ,  ( C `  I ) ,  if ( J  =  I ,  ( 1r `  R ) ,  .0.  ) ) )
 
Theoremmulmarep1el 19595 Element by element multiplication of a matrix with an identity matrix with a column replaced by a vector. (Contributed by AV, 16-Feb-2019.) (Revised by AV, 26-Feb-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  V  =  ( ( Base `  R )  ^m  N )   &    |-  .1.  =  ( 1r `  A )   &    |- 
 .0.  =  ( 0g `  R )   &    |-  E  =  ( (  .1.  ( N matRepV  R ) C ) `
  K )   =>    |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N )  /\  ( I  e.  N  /\  J  e.  N  /\  L  e.  N ) )  ->  ( ( I X L ) ( .r
 `  R ) ( L E J ) )  =  if ( J  =  K ,  ( ( I X L ) ( .r
 `  R ) ( C `  L ) ) ,  if ( J  =  L ,  ( I X L ) ,  .0.  ) ) )
 
Theoremmulmarep1gsum1 19596* The sum of element by element multiplications of a matrix with an identity matrix with a column replaced by a vector. (Contributed by AV, 16-Feb-2019.) (Revised by AV, 26-Feb-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  V  =  ( ( Base `  R )  ^m  N )   &    |-  .1.  =  ( 1r `  A )   &    |- 
 .0.  =  ( 0g `  R )   &    |-  E  =  ( (  .1.  ( N matRepV  R ) C ) `
  K )   =>    |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N )  /\  ( I  e.  N  /\  J  e.  N  /\  J  =/=  K ) )  ->  ( R  gsumg  ( l  e.  N  |->  ( ( I X l ) ( .r
 `  R ) ( l E J ) ) ) )  =  ( I X J ) )
 
Theoremmulmarep1gsum2 19597* The sum of element by element multiplications of a matrix with an identity matrix with a column replaced by a vector. (Contributed by AV, 18-Feb-2019.) (Revised by AV, 26-Feb-2019.)
 |-  A  =  ( N Mat 
 R )   &    |-  B  =  (
 Base `  A )   &    |-  V  =  ( ( Base `  R )  ^m  N )   &    |-  .1.  =  ( 1r `  A )   &    |- 
 .0.  =  ( 0g `  R )   &    |-  E  =  ( (  .1.  ( N matRepV  R ) C ) `
  K )   &    |-  .X.  =  ( R maVecMul  <. N ,  N >. )   =>    |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  C  e.  V  /\  K  e.  N )  /\  ( I  e.  N  /\  J  e.  N  /\  ( X  .X.  C )  =  Z ) ) 
 ->  ( R  gsumg  ( l  e.  N  |->  ( ( I X l ) ( .r
 `  R ) ( l E J ) ) ) )  =  if ( J  =  K ,  ( Z `  I ) ,  ( I X J ) ) )
 
Theorem1marepvmarrepid 19598 Replacing the ith row by 0's and the ith component of a (column) vector at the diagonal position for the identity matrix with the ith column replaced by the vector results in the matrix itself. (Contributed by AV, 14-Feb-2019.) (Revised by AV, 27-Feb-2019.)
 |-  V  =  ( (
 Base `  R )  ^m  N )   &    |-  .1.  =  ( 1r `  ( N Mat 
 R ) )   &    |-  X  =  ( (  .1.  ( N matRepV  R ) Z ) `
  I )   =>    |-  ( ( ( R  e.  Ring  /\  N  e.  Fin )  /\  ( I  e.  N  /\  Z  e.  V )
 )  ->  ( I
 ( X ( N matRRep  R ) ( Z `
  I ) ) I )  =  X )
 
11.2.8  Submatrices
 
Syntaxcsubma 19599 Syntax for submatrices of a square matrix.
 class subMat
 
Definitiondf-subma 19600* Define the submatrices of a square matrix. A submatrix is obtained by deleting a row and a column of the original matrix. Since the indices of a matrix need not to be sequential integers, it does not matter that there may be gaps in the numbering of the indices for the submatrix. The determinants of such submatrices are called the "minors" of the original matrix. (Contributed by AV, 27-Dec-2018.)
 |- subMat  =  ( n  e.  _V ,  r  e.  _V  |->  ( m  e.  ( Base `  ( n Mat  r
 ) )  |->  ( k  e.  n ,  l  e.  n  |->  ( i  e.  ( n  \  { k } ) ,  j  e.  ( n  \  { l }
 )  |->  ( i m j ) ) ) ) )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 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 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40161
  Copyright terms: Public domain < Previous  Next >