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Theorem List for Metamath Proof Explorer - 20101-20200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmat1dimmul 20101 The ring multiplication in the algebra of matrices with dimension 1. (Contributed by AV, 16-Aug-2019.) (Proof shortened by AV, 18-Apr-2021.)
𝐴 = ({𝐸} Mat 𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑂 = ⟨𝐸, 𝐸       (((𝑅 ∈ Ring ∧ 𝐸𝑉) ∧ (𝑋𝐵𝑌𝐵)) → ({⟨𝑂, 𝑋⟩} (.r𝐴){⟨𝑂, 𝑌⟩}) = {⟨𝑂, (𝑋(.r𝑅)𝑌)⟩})

Theoremmat1dimcrng 20102 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 ∈ CRing ∧ 𝐸𝑉) → 𝐴 ∈ CRing)

Theoremmat1f1o 20103* 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 = (𝑥𝐾 ↦ {⟨𝑂, 𝑥⟩})       ((𝑅 ∈ Ring ∧ 𝐸𝑉) → 𝐹:𝐾1-1-onto𝐵)

Theoremmat1rhmval 20104* 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 = (𝑥𝐾 ↦ {⟨𝑂, 𝑥⟩})       ((𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐾) → (𝐹𝑋) = {⟨𝑂, 𝑋⟩})

Theoremmat1rhmelval 20105* 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 = (𝑥𝐾 ↦ {⟨𝑂, 𝑥⟩})       ((𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐾) → (𝐸(𝐹𝑋)𝐸) = 𝑋)

Theoremmat1rhmcl 20106* 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 = (𝑥𝐾 ↦ {⟨𝑂, 𝑥⟩})       ((𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐾) → (𝐹𝑋) ∈ 𝐵)

Theoremmat1f 20107* 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 = (𝑥𝐾 ↦ {⟨𝑂, 𝑥⟩})       ((𝑅 ∈ Ring ∧ 𝐸𝑉) → 𝐹:𝐾𝐵)

Theoremmat1ghm 20108* 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 = (𝑥𝐾 ↦ {⟨𝑂, 𝑥⟩})       ((𝑅 ∈ Ring ∧ 𝐸𝑉) → 𝐹 ∈ (𝑅 GrpHom 𝐴))

Theoremmat1mhm 20109* 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 = (𝑥𝐾 ↦ {⟨𝑂, 𝑥⟩})    &   𝑀 = (mulGrp‘𝑅)    &   𝑁 = (mulGrp‘𝐴)       ((𝑅 ∈ Ring ∧ 𝐸𝑉) → 𝐹 ∈ (𝑀 MndHom 𝑁))

Theoremmat1rhm 20110* 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 = (𝑥𝐾 ↦ {⟨𝑂, 𝑥⟩})       ((𝑅 ∈ Ring ∧ 𝐸𝑉) → 𝐹 ∈ (𝑅 RingHom 𝐴))

Theoremmat1rngiso 20111* 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 = (𝑥𝐾 ↦ {⟨𝑂, 𝑥⟩})       ((𝑅 ∈ Ring ∧ 𝐸𝑉) → 𝐹 ∈ (𝑅 RingIso 𝐴))

Theoremmat1ric 20112 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 ∈ Ring ∧ 𝐸𝑉) → 𝑅𝑟 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 𝜆 𝐼 of the identity matrix 𝐼. Its effect on a vector is scalar multiplication by 𝜆 [see scmatscm 20138!]". 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 20115 and df-scmat 20116), basic properties of (elements of) these sets, and theorems showing that the diagonal matrices are a subring of the ring of square matrices (dmatsrng 20126), that the scalar matrices are a subring of the ring of square matrices (scmatsrng 20145), that the scalar matrices are a subring of the ring of diagonal matrices (scmatsrng1 20148) and that the ring of scalar matrices (over a commutative ring) is a commutative ring (scmatcrng 20146).

Syntaxcdmat 20113 Extend class notation for the algebra of diagonal matrices.
class DMat

Syntaxcscmat 20114 Extend class notation for the algebra of scalar matrices.
class ScMat

Definitiondf-dmat 20115* 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 ∈ Fin, 𝑟 ∈ V ↦ {𝑚 ∈ (Base‘(𝑛 Mat 𝑟)) ∣ ∀𝑖𝑛𝑗𝑛 (𝑖𝑗 → (𝑖𝑚𝑗) = (0g𝑟))})

Definitiondf-scmat 20116* 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 ∈ Fin, 𝑟 ∈ V ↦ (𝑛 Mat 𝑟) / 𝑎{𝑚 ∈ (Base‘𝑎) ∣ ∃𝑐 ∈ (Base‘𝑟)𝑚 = (𝑐( ·𝑠𝑎)(1r𝑎))})

Theoremdmatval 20117* 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 = (0g𝑅)    &   𝐷 = (𝑁 DMat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝐷 = {𝑚𝐵 ∣ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑚𝑗) = 0 )})

Theoremdmatel 20118* A 𝑁 x 𝑁 diagonal matrix over (a ring) 𝑅. (Contributed by AV, 18-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    0 = (0g𝑅)    &   𝐷 = (𝑁 DMat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑀𝐷 ↔ (𝑀𝐵 ∧ ∀𝑖𝑁𝑗𝑁 (𝑖𝑗 → (𝑖𝑀𝑗) = 0 ))))

Theoremdmatmat 20119 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 = (0g𝑅)    &   𝐷 = (𝑁 DMat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝑀𝐷𝑀𝐵))

Theoremdmatid 20120 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 = (0g𝑅)    &   𝐷 = (𝑁 DMat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r𝐴) ∈ 𝐷)

Theoremdmatelnd 20121 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 20122* 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 20123 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 20124 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 20125 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 20126 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 20127 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 20128 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 20129* 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 20130* 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 20131* 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 20132 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 20133 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 20134 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 20135* 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 20136* 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 20137* Alternate proof of scmate 20135: An entry of an 𝑁 x 𝑁 scalar matrix over the ring 𝑅. This prove makes use of scmatmats 20136 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 20138* 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 20139 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 20140 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 20141 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 20142 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 20143 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 20144 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 20145 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 20146 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 20147 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 20148 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 20149 Closure of the scalar multiplication in the ring of scalar matrices. (matvscl 20056 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 20150 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 20151 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 20152* 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 20153* 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 20154* 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 20155* 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 20156* 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 20157* 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 20158* 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 20159* 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 20160* 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 20161* 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 20162 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 20163 The empty matrix over a ring is a scalar matrix (and therefore, by scmatdmat 20140, also a diagonal matrix). (Contributed by AV, 21-Dec-2019.)
(𝑅 ∈ Ring → ∅ ∈ (∅ ScMat 𝑅))

Theoremmat1scmat 20164 A 1-dimensional matrix over a ring is always a scalar matrix (and therefore, by scmatdmat 20140, 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)))

11.2.6  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 20009 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 20166 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 19922. See also the statements in [Lang] p. 508.

Syntaxcmvmul 20165 Syntax for the operator for the multiplication of a vector with a matrix.
class maVecMul

Definitiondf-mvmul 20166* 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 20167* 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 20168* 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 20169* 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 20170* 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 20171* 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 20172 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 20173 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 20174 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 20175 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 20176 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 20177 Multiplication of a 0-dimensional matrix with a 0-dimensional vector. (Contributed by AV, 28-Feb-2019.)
· = (𝑅 maVecMul ⟨𝑁, 𝑁⟩)       ((𝑁 = ∅ ∧ 𝑅𝑉) → (∅ · ∅) = ∅)

Theoremmavmul0g 20178 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 20179* 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 20180* 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)∅)))

11.2.7  Replacement functions for a square matrix

Syntaxcmarrep 20181 Syntax for the row replacing function for a square matrix.
class matRRep

SyntaxcmatrepV 20182 Syntax for the function replacing a column of a matrix by a vector.
class matRepV

Definitiondf-marrep 20183* 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 20184* 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 20185* 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 = (0g𝑅)       𝑄 = (𝑚𝐵, 𝑠 ∈ (Base‘𝑅) ↦ (𝑘𝑁, 𝑙𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑘, if(𝑗 = 𝑙, 𝑠, 0 ), (𝑖𝑚𝑗)))))

Theoremmarrepval0 20186* 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 20187* 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 20188 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 20189 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 20190* 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 = (𝑚𝐵, 𝑣𝑉 ↦ (𝑘𝑁 ↦ (𝑖𝑁, 𝑗𝑁 ↦ if(𝑗 = 𝑘, (𝑣𝑖), (𝑖𝑚𝑗)))))

Theoremmarepvval0 20191* 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 20192* 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 20193 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 20194 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 20195 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 20196 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 20197 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 20198* 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 20199* 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 20200 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)

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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 42301-42360
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