Home Metamath Proof ExplorerTheorem List (p. 212 of 454) < Previous  Next > Bad symbols? Try the GIF version. Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

 Color key: Metamath Proof Explorer (1-28701) Hilbert Space Explorer (28702-30224) Users' Mathboxes (30225-45331)

Theorem List for Metamath Proof Explorer - 21101-21200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremdmatelnd 21101 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 = (0g𝑅)    &   𝐷 = (𝑁 DMat 𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋𝐷) ∧ (𝐼𝑁𝐽𝑁𝐼𝐽)) → (𝐼𝑋𝐽) = 0 )

Theoremdmatmul 21102* 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 = (0g𝑅)    &   𝐷 = (𝑁 DMat 𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → (𝑋(.r𝐴)𝑌) = (𝑥𝑁, 𝑦𝑁 ↦ if(𝑥 = 𝑦, ((𝑥𝑋𝑦)(.r𝑅)(𝑥𝑌𝑦)), 0 )))

Theoremdmatsubcl 21103 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 = (0g𝑅)    &   𝐷 = (𝑁 DMat 𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → (𝑋(-g𝐴)𝑌) ∈ 𝐷)

Theoremdmatsgrp 21104 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 = (0g𝑅)    &   𝐷 = (𝑁 DMat 𝑅)       ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 𝐷 ∈ (SubGrp‘𝐴))

Theoremdmatmulcl 21105 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 = (0g𝑅)    &   𝐷 = (𝑁 DMat 𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝐷𝑌𝐷)) → (𝑋(.r𝐴)𝑌) ∈ 𝐷)

Theoremdmatsrng 21106 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 = (0g𝑅)    &   𝐷 = (𝑁 DMat 𝑅)       ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 𝐷 ∈ (SubRing‘𝐴))

Theoremdmatcrng 21107 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 = (0g𝑅)    &   𝐷 = (𝑁 DMat 𝑅)    &   𝐶 = (𝐴s 𝐷)       ((𝑅 ∈ CRing ∧ 𝑁 ∈ Fin) → 𝐶 ∈ CRing)

Theoremdmatscmcl 21108 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)    &    = ( ·𝑠𝐴)    &   𝐷 = (𝑁 DMat 𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐶𝐾𝑀𝐷)) → (𝐶 𝑀) ∈ 𝐷)

Theoremscmatval 21109* 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 = (1r𝐴)    &    · = ( ·𝑠𝐴)    &   𝑆 = (𝑁 ScMat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑆 = {𝑚𝐵 ∣ ∃𝑐𝐾 𝑚 = (𝑐 · 1 )})

Theoremscmatel 21110* 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 = (1r𝐴)    &    · = ( ·𝑠𝐴)    &   𝑆 = (𝑁 ScMat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑀𝑆 ↔ (𝑀𝐵 ∧ ∃𝑐𝐾 𝑀 = (𝑐 · 1 ))))

Theoremscmatscmid 21111* 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 = (1r𝐴)    &    · = ( ·𝑠𝐴)    &   𝑆 = (𝑁 ScMat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅𝑉𝑀𝑆) → ∃𝑐𝐾 𝑀 = (𝑐 · 1 ))

Theoremscmatscmide 21112 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 = (0g𝑅)    &    1 = (1r𝐴)    &    = ( ·𝑠𝐴)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐶𝐵) ∧ (𝐼𝑁𝐽𝑁)) → (𝐼(𝐶 1 )𝐽) = if(𝐼 = 𝐽, 𝐶, 0 ))

Theoremscmatscmiddistr 21113 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 = (0g𝑅)    &    1 = (1r𝐴)    &    = ( ·𝑠𝐴)    &    · = (.r𝑅)    &    × = (.r𝐴)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑆𝐵𝑇𝐵)) → ((𝑆 1 ) × (𝑇 1 )) = ((𝑆 · 𝑇) 1 ))

Theoremscmatmat 21114 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 ∈ Fin ∧ 𝑅𝑉) → (𝑀𝑆𝑀𝐵))

Theoremscmate 21115* 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 = (0g𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝐼𝑁𝐽𝑁)) → ∃𝑐𝐾 (𝐼𝑀𝐽) = if(𝐼 = 𝐽, 𝑐, 0 ))

Theoremscmatmats 21116* 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 = (0g𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 = {𝑚𝐵 ∣ ∃𝑐𝐾𝑖𝑁𝑗𝑁 (𝑖𝑚𝑗) = if(𝑖 = 𝑗, 𝑐, 0 )})

TheoremscmateALT 21117* Alternate proof of scmate 21115: An entry of an 𝑁 x 𝑁 scalar matrix over the ring 𝑅. This prove makes use of scmatmats 21116 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 = (0g𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑀𝑆) ∧ (𝐼𝑁𝐽𝑁)) → ∃𝑐𝐾 (𝐼𝑀𝐽) = if(𝐼 = 𝐽, 𝑐, 0 ))

Theoremscmatscm 21118* 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)    &    = ( ·𝑠𝐴)    &    × = (.r𝐴)    &   𝑆 = (𝑁 ScMat 𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ 𝐶𝑆) → ∃𝑐𝐾𝑚𝐵 (𝐶 × 𝑚) = (𝑐 𝑚))

Theoremscmatid 21119 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 = (0g𝑅)    &   𝑆 = (𝑁 ScMat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r𝐴) ∈ 𝑆)

Theoremscmatdmat 21120 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 = (0g𝑅)    &   𝑆 = (𝑁 ScMat 𝑅)    &   𝐷 = (𝑁 DMat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (𝑀𝑆𝑀𝐷))

Theoremscmataddcl 21121 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 = (0g𝑅)    &   𝑆 = (𝑁 ScMat 𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝑆𝑌𝑆)) → (𝑋(+g𝐴)𝑌) ∈ 𝑆)

Theoremscmatsubcl 21122 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 = (0g𝑅)    &   𝑆 = (𝑁 ScMat 𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝑆𝑌𝑆)) → (𝑋(-g𝐴)𝑌) ∈ 𝑆)

Theoremscmatmulcl 21123 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 = (0g𝑅)    &   𝑆 = (𝑁 ScMat 𝑅)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝑋𝑆𝑌𝑆)) → (𝑋(.r𝐴)𝑌) ∈ 𝑆)

Theoremscmatsgrp 21124 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 = (0g𝑅)    &   𝑆 = (𝑁 ScMat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubGrp‘𝐴))

Theoremscmatsrng 21125 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 = (0g𝑅)    &   𝑆 = (𝑁 ScMat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubRing‘𝐴))

Theoremscmatcrng 21126 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 = (0g𝑅)    &   𝑆 = (𝑁 ScMat 𝑅)    &   𝐶 = (𝐴s 𝑆)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐶 ∈ CRing)

Theoremscmatsgrp1 21127 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 = (0g𝑅)    &   𝑆 = (𝑁 ScMat 𝑅)    &   𝐷 = (𝑁 DMat 𝑅)    &   𝐶 = (𝐴s 𝐷)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubGrp‘𝐶))

Theoremscmatsrng1 21128 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 = (0g𝑅)    &   𝑆 = (𝑁 ScMat 𝑅)    &   𝐷 = (𝑁 DMat 𝑅)    &   𝐶 = (𝐴s 𝐷)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (SubRing‘𝐶))

Theoremsmatvscl 21129 Closure of the scalar multiplication in the ring of scalar matrices. (matvscl 21036 analog.) (Contributed by AV, 24-Dec-2019.)
𝐾 = (Base‘𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝑆 = (𝑁 ScMat 𝑅)    &    = ( ·𝑠𝐴)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐶𝐾𝑋𝑆)) → (𝐶 𝑋) ∈ 𝑆)

Theoremscmatlss 21130 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 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝑆 ∈ (LSubSp‘𝐴))

Theoremscmatstrbas 21131 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 ∈ Fin ∧ 𝑅 ∈ Ring) → (Base‘𝑆) = 𝐶)

Theoremscmatrhmval 21132* The value of the ring homomorphism 𝐹. (Contributed by AV, 22-Dec-2019.)
𝐾 = (Base‘𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &    1 = (1r𝐴)    &    = ( ·𝑠𝐴)    &   𝐹 = (𝑥𝐾 ↦ (𝑥 1 ))       ((𝑅𝑉𝑋𝐾) → (𝐹𝑋) = (𝑋 1 ))

Theoremscmatrhmcl 21133* 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 = (1r𝐴)    &    = ( ·𝑠𝐴)    &   𝐹 = (𝑥𝐾 ↦ (𝑥 1 ))    &   𝐶 = (𝑁 ScMat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝑋𝐾) → (𝐹𝑋) ∈ 𝐶)

Theoremscmatf 21134* 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 = (1r𝐴)    &    = ( ·𝑠𝐴)    &   𝐹 = (𝑥𝐾 ↦ (𝑥 1 ))    &   𝐶 = (𝑁 ScMat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐹:𝐾𝐶)

Theoremscmatfo 21135* 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 = (1r𝐴)    &    = ( ·𝑠𝐴)    &   𝐹 = (𝑥𝐾 ↦ (𝑥 1 ))    &   𝐶 = (𝑁 ScMat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐹:𝐾onto𝐶)

Theoremscmatf1 21136* 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 = (1r𝐴)    &    = ( ·𝑠𝐴)    &   𝐹 = (𝑥𝐾 ↦ (𝑥 1 ))    &   𝐶 = (𝑁 ScMat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) → 𝐹:𝐾1-1𝐶)

Theoremscmatf1o 21137* 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 = (1r𝐴)    &    = ( ·𝑠𝐴)    &   𝐹 = (𝑥𝐾 ↦ (𝑥 1 ))    &   𝐶 = (𝑁 ScMat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) → 𝐹:𝐾1-1-onto𝐶)

Theoremscmatghm 21138* 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 = (1r𝐴)    &    = ( ·𝑠𝐴)    &   𝐹 = (𝑥𝐾 ↦ (𝑥 1 ))    &   𝐶 = (𝑁 ScMat 𝑅)    &   𝑆 = (𝐴s 𝐶)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐹 ∈ (𝑅 GrpHom 𝑆))

Theoremscmatmhm 21139* 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 = (1r𝐴)    &    = ( ·𝑠𝐴)    &   𝐹 = (𝑥𝐾 ↦ (𝑥 1 ))    &   𝐶 = (𝑁 ScMat 𝑅)    &   𝑆 = (𝐴s 𝐶)    &   𝑀 = (mulGrp‘𝑅)    &   𝑇 = (mulGrp‘𝑆)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐹 ∈ (𝑀 MndHom 𝑇))

Theoremscmatrhm 21140* 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 = (1r𝐴)    &    = ( ·𝑠𝐴)    &   𝐹 = (𝑥𝐾 ↦ (𝑥 1 ))    &   𝐶 = (𝑁 ScMat 𝑅)    &   𝑆 = (𝐴s 𝐶)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐹 ∈ (𝑅 RingHom 𝑆))

Theoremscmatrngiso 21141* 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 = (1r𝐴)    &    = ( ·𝑠𝐴)    &   𝐹 = (𝑥𝐾 ↦ (𝑥 1 ))    &   𝐶 = (𝑁 ScMat 𝑅)    &   𝑆 = (𝐴s 𝐶)       ((𝑁 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) → 𝐹 ∈ (𝑅 RingIso 𝑆))

Theoremscmatric 21142 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 ∈ Fin ∧ 𝑁 ≠ ∅ ∧ 𝑅 ∈ Ring) → 𝑅𝑟 𝑆)

Theoremmat0scmat 21143 The empty matrix over a ring is a scalar matrix (and therefore, by scmatdmat 21120, also a diagonal matrix). (Contributed by AV, 21-Dec-2019.)
(𝑅 ∈ Ring → ∅ ∈ (∅ ScMat 𝑅))

Theoremmat1scmat 21144 A 1-dimensional matrix over a ring is always a scalar matrix (and therefore, by scmatdmat 21120, also a diagonal matrix). (Contributed by AV, 21-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)       ((𝑁𝑉 ∧ (♯‘𝑁) = 1 ∧ 𝑅 ∈ Ring) → (𝑀𝐵𝑀 ∈ (𝑁 ScMat 𝑅)))

11.4.6  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 20991 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 21146 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 20448. See also the statements in [Lang] p. 508.

Syntaxcmvmul 21145 Syntax for the operator for the multiplication of a vector with a matrix.
class maVecMul

Definitiondf-mvmul 21146* The operator which multiplies an M x N -matrix with an N-dimensional vector. (Contributed by AV, 23-Feb-2019.)
maVecMul = (𝑟 ∈ V, 𝑜 ∈ V ↦ (1st𝑜) / 𝑚(2nd𝑜) / 𝑛(𝑥 ∈ ((Base‘𝑟) ↑m (𝑚 × 𝑛)), 𝑦 ∈ ((Base‘𝑟) ↑m 𝑛) ↦ (𝑖𝑚 ↦ (𝑟 Σg (𝑗𝑛 ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑦𝑗)))))))

Theoremmvmulfval 21147* Functional value of the matrix vector multiplication operator. (Contributed by AV, 23-Feb-2019.)
× = (𝑅 maVecMul ⟨𝑀, 𝑁⟩)    &   𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅𝑉)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)       (𝜑× = (𝑥 ∈ (𝐵m (𝑀 × 𝑁)), 𝑦 ∈ (𝐵m 𝑁) ↦ (𝑖𝑀 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑦𝑗)))))))

Theoremmvmulval 21148* Multiplication of a vector with a matrix. (Contributed by AV, 23-Feb-2019.)
× = (𝑅 maVecMul ⟨𝑀, 𝑁⟩)    &   𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅𝑉)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑋 ∈ (𝐵m (𝑀 × 𝑁)))    &   (𝜑𝑌 ∈ (𝐵m 𝑁))       (𝜑 → (𝑋 × 𝑌) = (𝑖𝑀 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌𝑗))))))

Theoremmvmulfv 21149* A cell/element in the vector resulting from a multiplication of a vector with a matrix. (Contributed by AV, 23-Feb-2019.)
× = (𝑅 maVecMul ⟨𝑀, 𝑁⟩)    &   𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅𝑉)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑋 ∈ (𝐵m (𝑀 × 𝑁)))    &   (𝜑𝑌 ∈ (𝐵m 𝑁))    &   (𝜑𝐼𝑀)       (𝜑 → ((𝑋 × 𝑌)‘𝐼) = (𝑅 Σg (𝑗𝑁 ↦ ((𝐼𝑋𝑗) · (𝑌𝑗)))))

Theoremmavmulval 21150* Multiplication of a vector with a square matrix. (Contributed by AV, 23-Feb-2019.)
𝐴 = (𝑁 Mat 𝑅)    &    × = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)    &   𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅𝑉)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑋 ∈ (Base‘𝐴))    &   (𝜑𝑌 ∈ (𝐵m 𝑁))       (𝜑 → (𝑋 × 𝑌) = (𝑖𝑁 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑋𝑗) · (𝑌𝑗))))))

Theoremmavmulfv 21151* 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)    &    × = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)    &   𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅𝑉)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑋 ∈ (Base‘𝐴))    &   (𝜑𝑌 ∈ (𝐵m 𝑁))    &   (𝜑𝐼𝑁)       (𝜑 → ((𝑋 × 𝑌)‘𝐼) = (𝑅 Σg (𝑗𝑁 ↦ ((𝐼𝑋𝑗) · (𝑌𝑗)))))

Theoremmavmulcl 21152 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)    &    × = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)    &   𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑋 ∈ (Base‘𝐴))    &   (𝜑𝑌 ∈ (𝐵m 𝑁))       (𝜑 → (𝑋 × 𝑌) ∈ (𝐵m 𝑁))

Theorem1mavmul 21153 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)    &    · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑌 ∈ (𝐵m 𝑁))       (𝜑 → ((1r𝐴) · 𝑌) = 𝑌)

Theoremmavmulass 21154 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)    &    · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑌 ∈ (𝐵m 𝑁))    &    × = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)    &   (𝜑𝑋 ∈ (Base‘𝐴))    &   (𝜑𝑍 ∈ (Base‘𝐴))       (𝜑 → ((𝑋 × 𝑍) · 𝑌) = (𝑋 · (𝑍 · 𝑌)))

Theoremmavmuldm 21155 The domain of the matrix vector multiplication function. (Contributed by AV, 27-Feb-2019.)
𝐵 = (Base‘𝑅)    &   𝐶 = (𝐵m (𝑀 × 𝑁))    &   𝐷 = (𝐵m 𝑁)    &    · = (𝑅 maVecMul ⟨𝑀, 𝑁⟩)       ((𝑅𝑉𝑀 ∈ Fin ∧ 𝑁 ∈ Fin) → dom · = (𝐶 × 𝐷))

Theoremmavmulsolcl 21156 Every solution of the equation 𝐴𝑋 = 𝑌 for a matrix 𝐴 and a vector 𝐵 is a vector. (Contributed by AV, 27-Feb-2019.)
𝐵 = (Base‘𝑅)    &   𝐶 = (𝐵m (𝑀 × 𝑁))    &   𝐷 = (𝐵m 𝑁)    &    · = (𝑅 maVecMul ⟨𝑀, 𝑁⟩)    &   𝐸 = (𝐵m 𝑀)       (((𝑀 ∈ Fin ∧ 𝑁 ∈ Fin ∧ 𝑀 ≠ ∅) ∧ (𝑅𝑉𝑌𝐸)) → ((𝐴 · 𝑋) = 𝑌𝑋𝐷))

Theoremmavmul0 21157 Multiplication of a 0-dimensional matrix with a 0-dimensional vector. (Contributed by AV, 28-Feb-2019.)
· = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)       ((𝑁 = ∅ ∧ 𝑅𝑉) → (∅ · ∅) = ∅)

Theoremmavmul0g 21158 The result of the 0-dimensional multiplication of a matrix with a vector is always the empty set. (Contributed by AV, 1-Mar-2019.)
· = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)       ((𝑁 = ∅ ∧ 𝑅𝑉) → (𝑋 · 𝑌) = ∅)

Theoremmvmumamul1 21159* 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.)
× = (𝑅 maMul ⟨𝑀, 𝑁, {∅}⟩)    &    · = (𝑅 maVecMul ⟨𝑀, 𝑁⟩)    &   𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝐴 ∈ (𝐵m (𝑀 × 𝑁)))    &   (𝜑𝑌 ∈ (𝐵m 𝑁))    &   (𝜑𝑍 ∈ (𝐵m (𝑁 × {∅})))       (𝜑 → (∀𝑗𝑁 (𝑌𝑗) = (𝑗𝑍∅) → ∀𝑖𝑀 ((𝐴 · 𝑌)‘𝑖) = (𝑖(𝐴 × 𝑍)∅)))

Theoremmavmumamul1 21160* 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)    &    × = (𝑅 maMul ⟨𝑁, 𝑁, {∅}⟩)    &    · = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)    &   𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑋 ∈ (Base‘𝐴))    &   (𝜑𝑌 ∈ (𝐵m 𝑁))    &   (𝜑𝑍 ∈ (𝐵m (𝑁 × {∅})))       (𝜑 → (∀𝑗𝑁 (𝑌𝑗) = (𝑗𝑍∅) → ∀𝑖𝑁 ((𝑋 · 𝑌)‘𝑖) = (𝑖(𝑋 × 𝑍)∅)))

11.4.7  Replacement functions for a square matrix

Syntaxcmarrep 21161 Syntax for the row replacing function for a square matrix.
class matRRep

SyntaxcmatrepV 21162 Syntax for the function replacing a column of a matrix by a vector.
class matRepV

Definitiondf-marrep 21163* 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 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)), 𝑠 ∈ (Base‘𝑟) ↦ (𝑘𝑛, 𝑙𝑛 ↦ (𝑖𝑛, 𝑗𝑛 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, (0g𝑟)), (𝑖𝑚𝑗))))))

Definitiondf-marepv 21164* Function replacing a column of a matrix by a vector. (Contributed by AV, 9-Feb-2019.) (Revised by AV, 26-Feb-2019.)
matRepV = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)), 𝑣 ∈ ((Base‘𝑟) ↑m 𝑛) ↦ (𝑘𝑛 ↦ (𝑖𝑛, 𝑗𝑛 ↦ if(𝑗 = 𝑘, (𝑣𝑖), (𝑖𝑚𝑗))))))

Theoremmarrepfval 21165* First substitution for the definition of the matrix row replacement function. (Contributed by AV, 12-Feb-2019.) (Proof shortened by AV, 2-Mar-2024.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑄 = (𝑁 matRRep 𝑅)    &    0 = (0g𝑅)       𝑄 = (𝑚𝐵, 𝑠 ∈ (Base‘𝑅) ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗)))))

Theoremmarrepval0 21166* 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 = (0g𝑅)       ((𝑀𝐵𝑆 ∈ (Base‘𝑅)) → (𝑀𝑄𝑆) = (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑆, 0 ), (𝑖𝑀𝑗)))))

Theoremmarrepval 21167* 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 = (0g𝑅)       (((𝑀𝐵𝑆 ∈ (Base‘𝑅)) ∧ (𝐾𝑁𝐿𝑁)) → (𝐾(𝑀𝑄𝑆)𝐿) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝐾, if(𝑗 = 𝐿, 𝑆, 0 ), (𝑖𝑀𝑗))))

Theoremmarrepeval 21168 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 = (0g𝑅)       (((𝑀𝐵𝑆 ∈ (Base‘𝑅)) ∧ (𝐾𝑁𝐿𝑁) ∧ (𝐼𝑁𝐽𝑁)) → (𝐼(𝐾(𝑀𝑄𝑆)𝐿)𝐽) = if(𝐼 = 𝐾, if(𝐽 = 𝐿, 𝑆, 0 ), (𝐼𝑀𝐽)))

Theoremmarrepcl 21169 Closure of the row replacement function for square matrices. (Contributed by AV, 13-Feb-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)       (((𝑅 ∈ Ring ∧ 𝑀𝐵𝑆 ∈ (Base‘𝑅)) ∧ (𝐾𝑁𝐿𝑁)) → (𝐾(𝑀(𝑁 matRRep 𝑅)𝑆)𝐿) ∈ 𝐵)

Theoremmarepvfval 21170* 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.) (Proof shortened by AV, 2-Mar-2024.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑄 = (𝑁 matRepV 𝑅)    &   𝑉 = ((Base‘𝑅) ↑m 𝑁)       𝑄 = (𝑚𝐵, 𝑣𝑉 ↦ (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝑣𝑖), (𝑖𝑚𝑗)))))

Theoremmarepvval0 21171* 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)       ((𝑀𝐵𝐶𝑉) → (𝑀𝑄𝐶) = (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝐶𝑖), (𝑖𝑀𝑗)))))

Theoremmarepvval 21172* 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)       ((𝑀𝐵𝐶𝑉𝐾𝑁) → ((𝑀𝑄𝐶)‘𝐾) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝐾, (𝐶𝑖), (𝑖𝑀𝑗))))

Theoremmarepveval 21173 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)       (((𝑀𝐵𝐶𝑉𝐾𝑁) ∧ (𝐼𝑁𝐽𝑁)) → (𝐼((𝑀𝑄𝐶)‘𝐾)𝐽) = if(𝐽 = 𝐾, (𝐶𝐼), (𝐼𝑀𝐽)))

Theoremmarepvcl 21174 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 ∈ Ring ∧ (𝑀𝐵𝐶𝑉𝐾𝑁)) → ((𝑀(𝑁 matRepV 𝑅)𝐶)‘𝐾) ∈ 𝐵)

Theoremma1repvcl 21175 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 = (1r𝐴)       (((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐶𝑉𝐾𝑁)) → (( 1 (𝑁 matRepV 𝑅)𝐶)‘𝐾) ∈ 𝐵)

Theoremma1repveval 21176 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 = (1r𝐴)    &    0 = (0g𝑅)    &   𝐸 = (( 1 (𝑁 matRepV 𝑅)𝐶)‘𝐾)       ((𝑅 ∈ Ring ∧ (𝑀𝐵𝐶𝑉𝐾𝑁) ∧ (𝐼𝑁𝐽𝑁)) → (𝐼𝐸𝐽) = if(𝐽 = 𝐾, (𝐶𝐼), if(𝐽 = 𝐼, (1r𝑅), 0 )))

Theoremmulmarep1el 21177 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 = (1r𝐴)    &    0 = (0g𝑅)    &   𝐸 = (( 1 (𝑁 matRepV 𝑅)𝐶)‘𝐾)       ((𝑅 ∈ Ring ∧ (𝑋𝐵𝐶𝑉𝐾𝑁) ∧ (𝐼𝑁𝐽𝑁𝐿𝑁)) → ((𝐼𝑋𝐿)(.r𝑅)(𝐿𝐸𝐽)) = if(𝐽 = 𝐾, ((𝐼𝑋𝐿)(.r𝑅)(𝐶𝐿)), if(𝐽 = 𝐿, (𝐼𝑋𝐿), 0 )))

Theoremmulmarep1gsum1 21178* 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 = (1r𝐴)    &    0 = (0g𝑅)    &   𝐸 = (( 1 (𝑁 matRepV 𝑅)𝐶)‘𝐾)       ((𝑅 ∈ Ring ∧ (𝑋𝐵𝐶𝑉𝐾𝑁) ∧ (𝐼𝑁𝐽𝑁𝐽𝐾)) → (𝑅 Σg (𝑙𝑁 ↦ ((𝐼𝑋𝑙)(.r𝑅)(𝑙𝐸𝐽)))) = (𝐼𝑋𝐽))

Theoremmulmarep1gsum2 21179* 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 = (1r𝐴)    &    0 = (0g𝑅)    &   𝐸 = (( 1 (𝑁 matRepV 𝑅)𝐶)‘𝐾)    &    × = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)       ((𝑅 ∈ Ring ∧ (𝑋𝐵𝐶𝑉𝐾𝑁) ∧ (𝐼𝑁𝐽𝑁 ∧ (𝑋 × 𝐶) = 𝑍)) → (𝑅 Σg (𝑙𝑁 ↦ ((𝐼𝑋𝑙)(.r𝑅)(𝑙𝐸𝐽)))) = if(𝐽 = 𝐾, (𝑍𝐼), (𝐼𝑋𝐽)))

Theorem1marepvmarrepid 21180 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 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → (𝐼(𝑋(𝑁 matRRep 𝑅)(𝑍𝐼))𝐼) = 𝑋)

11.4.8  Submatrices

Syntaxcsubma 21181 Syntax for submatrices of a square matrix.
class subMat

Definitiondf-subma 21182* 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 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑘𝑛, 𝑙𝑛 ↦ (𝑖 ∈ (𝑛 ∖ {𝑘}), 𝑗 ∈ (𝑛 ∖ {𝑙}) ↦ (𝑖𝑚𝑗)))))

Theoremsubmabas 21183* Any subset of the index set of a square matrix defines a submatrix of the matrix. (Contributed by AV, 1-Jan-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)       ((𝑀𝐵𝐷𝑁) → (𝑖𝐷, 𝑗𝐷 ↦ (𝑖𝑀𝑗)) ∈ (Base‘(𝐷 Mat 𝑅)))

Theoremsubmafval 21184* First substitution for a submatrix. (Contributed by AV, 28-Dec-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (𝑁 subMat 𝑅)    &   𝐵 = (Base‘𝐴)       𝑄 = (𝑚𝐵 ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑚𝑗))))

Theoremsubmaval0 21185* Second substitution for a submatrix. (Contributed by AV, 28-Dec-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (𝑁 subMat 𝑅)    &   𝐵 = (Base‘𝐴)       (𝑀𝐵 → (𝑄𝑀) = (𝑘𝑁, 𝑙𝑁 ↦ (𝑖 ∈ (𝑁 ∖ {𝑘}), 𝑗 ∈ (𝑁 ∖ {𝑙}) ↦ (𝑖𝑀𝑗))))

Theoremsubmaval 21186* Third substitution for a submatrix. (Contributed by AV, 28-Dec-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (𝑁 subMat 𝑅)    &   𝐵 = (Base‘𝐴)       ((𝑀𝐵𝐾𝑁𝐿𝑁) → (𝐾(𝑄𝑀)𝐿) = (𝑖 ∈ (𝑁 ∖ {𝐾}), 𝑗 ∈ (𝑁 ∖ {𝐿}) ↦ (𝑖𝑀𝑗)))

Theoremsubmaeval 21187 An entry of a submatrix of a square matrix. (Contributed by AV, 28-Dec-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝑄 = (𝑁 subMat 𝑅)    &   𝐵 = (Base‘𝐴)       ((𝑀𝐵 ∧ (𝐾𝑁𝐿𝑁) ∧ (𝐼 ∈ (𝑁 ∖ {𝐾}) ∧ 𝐽 ∈ (𝑁 ∖ {𝐿}))) → (𝐼(𝐾(𝑄𝑀)𝐿)𝐽) = (𝐼𝑀𝐽))

Theorem1marepvsma1 21188 The submatrix of the identity matrix with the ith column replaced by the vector obtained by removing the ith row and the ith column is an identity matrix. (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 ∈ Ring ∧ 𝑁 ∈ Fin) ∧ (𝐼𝑁𝑍𝑉)) → (𝐼((𝑁 subMat 𝑅)‘𝑋)𝐼) = (1r‘((𝑁 ∖ {𝐼}) Mat 𝑅)))

11.5  The determinant

11.5.1  Definition and basic properties

Syntaxcmdat 21189 Syntax for the matrix determinant function.

Definitiondf-mdet 21190* Determinant of a square matrix. This definition is based on Leibniz' Formula (see mdetleib 21192). The properties of the axiomatic definition of a determinant according to [Weierstrass] p. 272 are derived from this definition as theorems: "The determinant function is the unique multilinear, alternating and normalized function from the algebra of square matrices of the same dimension over a commutative ring to this ring". The functionality is shown by mdetf 21200. Multilineary means "linear for each row" - the additivity is shown by mdetrlin 21207, the homogeneity by mdetrsca 21208. Furthermore, it is shown that the determinant function is alternating (see mdetralt 21213) and normalized (see mdet1 21206). Finally, the uniqueness is shown by mdetuni 21227. As a consequence, the "determinant of a square matrix" is the function value of the determinant function for this square matrix, see mdetleib 21192. (Contributed by Stefan O'Rear, 9-Sep-2015.) (Revised by SO, 10-Jul-2018.)
maDet = (𝑛 ∈ V, 𝑟 ∈ V ↦ (𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ↦ (𝑟 Σg (𝑝 ∈ (Base‘(SymGrp‘𝑛)) ↦ ((((ℤRHom‘𝑟) ∘ (pmSgn‘𝑛))‘𝑝)(.r𝑟)((mulGrp‘𝑟) Σg (𝑥𝑛 ↦ ((𝑝𝑥)𝑚𝑥))))))))

Theoremmdetfval 21191* First substitution for the determinant definition. (Contributed by Stefan O'Rear, 9-Sep-2015.) (Revised by SO, 9-Jul-2018.)
𝐷 = (𝑁 maDet 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑌 = (ℤRHom‘𝑅)    &   𝑆 = (pmSgn‘𝑁)    &    · = (.r𝑅)    &   𝑈 = (mulGrp‘𝑅)       𝐷 = (𝑚𝐵 ↦ (𝑅 Σg (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥)))))))

Theoremmdetleib 21192* Full substitution of our determinant definition (also known as Leibniz' Formula, expanding by columns). Proposition 4.6 in [Lang] p. 514. (Contributed by Stefan O'Rear, 3-Oct-2015.) (Revised by SO, 9-Jul-2018.)
𝐷 = (𝑁 maDet 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑌 = (ℤRHom‘𝑅)    &   𝑆 = (pmSgn‘𝑁)    &    · = (.r𝑅)    &   𝑈 = (mulGrp‘𝑅)       (𝑀𝐵 → (𝐷𝑀) = (𝑅 Σg (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑀𝑥)))))))

Theoremmdetleib2 21193* Leibniz' formula can also be expanded by rows. (Contributed by Stefan O'Rear, 9-Jul-2018.) (Proof shortened by AV, 23-Jul-2019.)
𝐷 = (𝑁 maDet 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑌 = (ℤRHom‘𝑅)    &   𝑆 = (pmSgn‘𝑁)    &    · = (.r𝑅)    &   𝑈 = (mulGrp‘𝑅)       ((𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝐷𝑀) = (𝑅 Σg (𝑝𝑃 ↦ (((𝑌𝑆)‘𝑝) · (𝑈 Σg (𝑥𝑁 ↦ (𝑥𝑀(𝑝𝑥))))))))

Theoremnfimdetndef 21194 The determinant is not defined for an infinite matrix. (Contributed by AV, 27-Dec-2018.)
𝐷 = (𝑁 maDet 𝑅)       (𝑁 ∉ Fin → 𝐷 = ∅)

Theoremmdetfval1 21195* First substitution of an alternative determinant definition. (Contributed by Stefan O'Rear, 9-Sep-2015.) (Revised by AV, 27-Dec-2018.)
𝐷 = (𝑁 maDet 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑌 = (ℤRHom‘𝑅)    &   𝑆 = (pmSgn‘𝑁)    &    · = (.r𝑅)    &   𝑈 = (mulGrp‘𝑅)       𝐷 = (𝑚𝐵 ↦ (𝑅 Σg (𝑝𝑃 ↦ ((𝑌‘(𝑆𝑝)) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑚𝑥)))))))

Theoremmdetleib1 21196* Full substitution of an alternative determinant definition (also known as Leibniz' Formula). (Contributed by Stefan O'Rear, 3-Oct-2015.) (Revised by AV, 26-Dec-2018.)
𝐷 = (𝑁 maDet 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑌 = (ℤRHom‘𝑅)    &   𝑆 = (pmSgn‘𝑁)    &    · = (.r𝑅)    &   𝑈 = (mulGrp‘𝑅)       (𝑀𝐵 → (𝐷𝑀) = (𝑅 Σg (𝑝𝑃 ↦ ((𝑌‘(𝑆𝑝)) · (𝑈 Σg (𝑥𝑁 ↦ ((𝑝𝑥)𝑀𝑥)))))))

Theoremmdet0pr 21197 The determinant for 0-dimensional matrices is a singleton containing an ordered pair with the singleton containing the empty set as first component, and the singleton containing the 1 element of the underlying ring as second component. (Contributed by AV, 28-Feb-2019.)
(𝑅 ∈ Ring → (∅ maDet 𝑅) = {⟨∅, (1r𝑅)⟩})

Theoremmdet0f1o 21198 The determinant for 0-dimensional matrices is a one-to-one function from the singleton containing the empty set onto the singleton containing the 1 element of the underlying ring.function x is . (Contributed by AV, 28-Feb-2019.)
(𝑅 ∈ Ring → (∅ maDet 𝑅):{∅}–1-1-onto→{(1r𝑅)})

Theoremmdet0fv0 21199 The determinant of a 0-dimensional matrix is the 1 element of the underlying ring . (Contributed by AV, 28-Feb-2019.)
(𝑅 ∈ Ring → ((∅ maDet 𝑅)‘∅) = (1r𝑅))

Theoremmdetf 21200 Functionality of the determinant, see also definition in [Lang] p. 513. (Contributed by Stefan O'Rear, 9-Jul-2018.) (Proof shortened by AV, 23-Jul-2019.)
𝐷 = (𝑁 maDet 𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐾 = (Base‘𝑅)       (𝑅 ∈ CRing → 𝐷:𝐵𝐾)

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 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45331
 Copyright terms: Public domain < Previous  Next >