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Theorem List for Metamath Proof Explorer - 20301-20400   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmat1 20301* Value of an identity matrix, see also the statement in [Lang] p. 504: "The unit element of the ring of n x n matrices is the matrix In ... whose components are equal to 0 except on the diagonal, in which case they are equal to 1.". (Contributed by Stefan O'Rear, 7-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &    1 = (1r𝑅)    &    0 = (0g𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (1r𝐴) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, 1 , 0 )))

Theoremmat1ov 20302 Entries of an identity matrix, deduction form. (Contributed by Stefan O'Rear, 10-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &    1 = (1r𝑅)    &    0 = (0g𝑅)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝐼𝑁)    &   (𝜑𝐽𝑁)    &   𝑈 = (1r𝐴)       (𝜑 → (𝐼𝑈𝐽) = if(𝐼 = 𝐽, 1 , 0 ))

Theoremmat1bas 20303 The identity matrix is a matrix. (Contributed by AV, 15-Feb-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    1 = (1r‘(𝑁 Mat 𝑅))       ((𝑅 ∈ Ring ∧ 𝑁 ∈ Fin) → 1𝐵)

Theoremmatsc 20304* The identity matrix multiplied with a scalar. (Contributed by Stefan O'Rear, 16-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐾 = (Base‘𝑅)    &    · = ( ·𝑠𝐴)    &    0 = (0g𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring ∧ 𝐿𝐾) → (𝐿 · (1r𝐴)) = (𝑖𝑁, 𝑗𝑁 ↦ if(𝑖 = 𝑗, 𝐿, 0 )))

Theoremofco2 20305 Distribution law for the function operation and the composition of functions. (Contributed by Stefan O'Rear, 17-Jul-2018.)
(((𝐹 ∈ V ∧ 𝐺 ∈ V) ∧ (Fun 𝐻 ∧ (𝐹𝐻) ∈ V ∧ (𝐺𝐻) ∈ V)) → ((𝐹𝑓 𝑅𝐺) ∘ 𝐻) = ((𝐹𝐻) ∘𝑓 𝑅(𝐺𝐻)))

Theoremoftpos 20306 The transposition of the value of a function operation for two functions is the value of the function operation for the two functions transposed. (Contributed by Stefan O'Rear, 17-Jul-2018.)
((𝐹𝑉𝐺𝑊) → tpos (𝐹𝑓 𝑅𝐺) = (tpos 𝐹𝑓 𝑅tpos 𝐺))

Theoremmattposcl 20307 The transpose of a square matrix is a square matrix of the same size. (Contributed by SO, 9-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)       (𝑀𝐵 → tpos 𝑀𝐵)

Theoremmattpostpos 20308 The transpose of the transpose of a square matrix is the square matrix itself. (Contributed by SO, 17-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)       (𝑀𝐵 → tpos tpos 𝑀 = 𝑀)

Theoremmattposvs 20309 The transposition of a matrix multiplied with a scalar equals the transposed matrix multiplied with the scalar, see also the statement in [Lang] p. 505. (Contributed by Stefan O'Rear, 17-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐾 = (Base‘𝑅)    &    · = ( ·𝑠𝐴)       ((𝑋𝐾𝑌𝐵) → tpos (𝑋 · 𝑌) = (𝑋 · tpos 𝑌))

Theoremmattpos1 20310 The transposition of the identity matrix is the identity matrix. (Contributed by Stefan O'Rear, 17-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &    1 = (1r𝐴)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → tpos 1 = 1 )

Theoremtposmap 20311 The transposition of an I X J -matrix is a J X I -matrix, see also the statement in [Lang] p. 505. (Contributed by Stefan O'Rear, 9-Jul-2018.)
(𝐴 ∈ (𝐵𝑚 (𝐼 × 𝐽)) → tpos 𝐴 ∈ (𝐵𝑚 (𝐽 × 𝐼)))

Theoremmamutpos 20312 Behavior of transposes in matrix products, see also the statement in [Lang] p. 505. (Contributed by Stefan O'Rear, 9-Jul-2018.)
𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑃⟩)    &   𝐺 = (𝑅 maMul ⟨𝑃, 𝑁, 𝑀⟩)    &   𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ CRing)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑃 ∈ Fin)    &   (𝜑𝑋 ∈ (𝐵𝑚 (𝑀 × 𝑁)))    &   (𝜑𝑌 ∈ (𝐵𝑚 (𝑁 × 𝑃)))       (𝜑 → tpos (𝑋𝐹𝑌) = (tpos 𝑌𝐺tpos 𝑋))

Theoremmattposm 20313 Multiplying two transposed matrices results in the transposition of the product of the two matrices. (Contributed by Stefan O'Rear, 17-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    · = (.r𝐴)       ((𝑅 ∈ CRing ∧ 𝑋𝐵𝑌𝐵) → tpos (𝑋 · 𝑌) = (tpos 𝑌 · tpos 𝑋))

Theoremmatgsumcl 20314* Closure of a group sum over the diagonal coefficients of a square matrix over a commutative ring. (Contributed by AV, 29-Dec-2018.) (Proof shortened by AV, 23-Jul-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑈 = (mulGrp‘𝑅)       ((𝑅 ∈ CRing ∧ 𝑀𝐵) → (𝑈 Σg (𝑟𝑁 ↦ (𝑟𝑀𝑟))) ∈ (Base‘𝑅))

Theoremmadetsumid 20315* The identity summand in the Leibniz' formula of a determinant for a square matrix over a commutative ring. (Contributed by AV, 29-Dec-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑈 = (mulGrp‘𝑅)    &   𝑌 = (ℤRHom‘𝑅)    &   𝑆 = (pmSgn‘𝑁)    &    · = (.r𝑅)       ((𝑅 ∈ CRing ∧ 𝑀𝐵𝑃 = ( I ↾ 𝑁)) → (((𝑌𝑆)‘𝑃) · (𝑈 Σg (𝑟𝑁 ↦ ((𝑃𝑟)𝑀𝑟)))) = (𝑈 Σg (𝑟𝑁 ↦ (𝑟𝑀𝑟))))

Theoremmatepmcl 20316* Each entry of a matrix with an index as permutation of the other is an element of the underlying ring. (Contributed by AV, 1-Jan-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Base‘(SymGrp‘𝑁))       ((𝑅 ∈ Ring ∧ 𝑄𝑃𝑀𝐵) → ∀𝑛𝑁 ((𝑄𝑛)𝑀𝑛) ∈ (Base‘𝑅))

Theoremmatepm2cl 20317* Each entry of a matrix with an index as permutation of the other is an element of the underlying ring. (Contributed by AV, 1-Jan-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑃 = (Base‘(SymGrp‘𝑁))       ((𝑅 ∈ Ring ∧ 𝑄𝑃𝑀𝐵) → ∀𝑛𝑁 (𝑛𝑀(𝑄𝑛)) ∈ (Base‘𝑅))

Theoremmadetsmelbas 20318* A summand of the determinant of a matrix belongs to the underlying ring. (Contributed by AV, 1-Jan-2019.)
𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑆 = (pmSgn‘𝑁)    &   𝑌 = (ℤRHom‘𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐺 = (mulGrp‘𝑅)       ((𝑅 ∈ CRing ∧ 𝑀𝐵𝑄𝑃) → (((𝑌𝑆)‘𝑄)(.r𝑅)(𝐺 Σg (𝑛𝑁 ↦ ((𝑄𝑛)𝑀𝑛)))) ∈ (Base‘𝑅))

Theoremmadetsmelbas2 20319* A summand of the determinant of a matrix belongs to the underlying ring. (Contributed by AV, 1-Jan-2019.)
𝑃 = (Base‘(SymGrp‘𝑁))    &   𝑆 = (pmSgn‘𝑁)    &   𝑌 = (ℤRHom‘𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐺 = (mulGrp‘𝑅)       ((𝑅 ∈ CRing ∧ 𝑀𝐵𝑄𝑃) → (((𝑌𝑆)‘𝑄)(.r𝑅)(𝐺 Σg (𝑛𝑁 ↦ (𝑛𝑀(𝑄𝑛))))) ∈ (Base‘𝑅))

11.2.4  Matrices of dimension 0 and 1

As already mentioned before, and shown in mat0dimbas0 20320, the empty set is the sole zero-dimensional matrix (also called "empty matrix", see Wikipedia https://en.wikipedia.org/wiki/Matrix_(mathematics)#Empty_matrices). In the following, some properties of the empty matrix are shown, especially that the empty matrix over an arbitrary ring forms a commutative ring, see mat0dimcrng 20324.

For the one-dimensional case, it can be shown that a ring of matrices with dimension 1 is isomorphic to the underlying ring, see mat1ric 20341.

Theoremmat0dimbas0 20320 The empty set is the one and only matrix of dimension 0, called "the empty matrix". (Contributed by AV, 27-Feb-2019.)
(𝑅𝑉 → (Base‘(∅ Mat 𝑅)) = {∅})

Theoremmat0dim0 20321 The zero of the algebra of matrices with dimension 0. (Contributed by AV, 6-Aug-2019.)
𝐴 = (∅ Mat 𝑅)       (𝑅 ∈ Ring → (0g𝐴) = ∅)

Theoremmat0dimid 20322 The identity of the algebra of matrices with dimension 0. (Contributed by AV, 6-Aug-2019.)
𝐴 = (∅ Mat 𝑅)       (𝑅 ∈ Ring → (1r𝐴) = ∅)

Theoremmat0dimscm 20323 The scalar multiplication in the algebra of matrices with dimension 0. (Contributed by AV, 6-Aug-2019.)
𝐴 = (∅ Mat 𝑅)       ((𝑅 ∈ Ring ∧ 𝑋 ∈ (Base‘𝑅)) → (𝑋( ·𝑠𝐴)∅) = ∅)

Theoremmat0dimcrng 20324 The algebra of matrices with dimension 0 (over an arbitrary ring!) is a commutative ring. (Contributed by AV, 10-Aug-2019.)
𝐴 = (∅ Mat 𝑅)       (𝑅 ∈ Ring → 𝐴 ∈ CRing)

Theoremmat1dimelbas 20325* A matrix with dimension 1 is an ordered pair with an ordered pair (of the one and only pair of indices) as first component. (Contributed by AV, 15-Aug-2019.)
𝐴 = ({𝐸} Mat 𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑂 = ⟨𝐸, 𝐸       ((𝑅 ∈ Ring ∧ 𝐸𝑉) → (𝑀 ∈ (Base‘𝐴) ↔ ∃𝑟𝐵 𝑀 = {⟨𝑂, 𝑟⟩}))

Theoremmat1dimbas 20326 A matrix with dimension 1 is an ordered pair with an ordered pair (of the one and only pair of indices) as first component. (Contributed by AV, 15-Aug-2019.)
𝐴 = ({𝐸} Mat 𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑂 = ⟨𝐸, 𝐸       ((𝑅 ∈ Ring ∧ 𝐸𝑉𝑋𝐵) → {⟨𝑂, 𝑋⟩} ∈ (Base‘𝐴))

Theoremmat1dim0 20327 The zero of the algebra of matrices with dimension 1. (Contributed by AV, 15-Aug-2019.)
𝐴 = ({𝐸} Mat 𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑂 = ⟨𝐸, 𝐸       ((𝑅 ∈ Ring ∧ 𝐸𝑉) → (0g𝐴) = {⟨𝑂, (0g𝑅)⟩})

Theoremmat1dimid 20328 The identity of the algebra of matrices with dimension 1. (Contributed by AV, 15-Aug-2019.)
𝐴 = ({𝐸} Mat 𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑂 = ⟨𝐸, 𝐸       ((𝑅 ∈ Ring ∧ 𝐸𝑉) → (1r𝐴) = {⟨𝑂, (1r𝑅)⟩})

Theoremmat1dimscm 20329 The scalar multiplication in the algebra of matrices with dimension 1. (Contributed by AV, 16-Aug-2019.)
𝐴 = ({𝐸} Mat 𝑅)    &   𝐵 = (Base‘𝑅)    &   𝑂 = ⟨𝐸, 𝐸       (((𝑅 ∈ Ring ∧ 𝐸𝑉) ∧ (𝑋𝐵𝑌𝐵)) → (𝑋( ·𝑠𝐴){⟨𝑂, 𝑌⟩}) = {⟨𝑂, (𝑋(.r𝑅)𝑌)⟩})

Theoremmat1dimmul 20330 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 20331 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 20332* 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 20333* 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 20334* 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 20335* 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 20336* 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 20337* 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 20338* 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 20339* 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 20340* 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 20341 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 20367!]". 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 20344 and df-scmat 20345), basic properties of (elements of) these sets, and theorems showing that the diagonal matrices are a subring of the ring of square matrices (dmatsrng 20355), that the scalar matrices are a subring of the ring of square matrices (scmatsrng 20374), that the scalar matrices are a subring of the ring of diagonal matrices (scmatsrng1 20377) and that the ring of scalar matrices (over a commutative ring) is a commutative ring (scmatcrng 20375).

Syntaxcdmat 20342 Extend class notation for the algebra of diagonal matrices.
class DMat

Syntaxcscmat 20343 Extend class notation for the algebra of scalar matrices.
class ScMat

Definitiondf-dmat 20344* 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 20345* 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 20346* 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 20347* 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 20348 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 20349 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 20350 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 20351* 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 20352 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 20353 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 20354 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 20355 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 20356 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 20357 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 20358* 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 20359* 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 20360* 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 20361 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 20362 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 20363 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 20364* 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 20365* 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 20366* Alternate proof of scmate 20364: An entry of an 𝑁 x 𝑁 scalar matrix over the ring 𝑅. This prove makes use of scmatmats 20365 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 20367* 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 20368 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 20369 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 20370 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 20371 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 20372 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 20373 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 20374 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 20375 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 20376 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 20377 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 20378 Closure of the scalar multiplication in the ring of scalar matrices. (matvscl 20285 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 20379 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 20380 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 20381* 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 20382* 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 20383* 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 20384* 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 20385* 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 20386* 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 20387* 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 20388* 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 20389* 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 20390* 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 20391 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 20392 The empty matrix over a ring is a scalar matrix (and therefore, by scmatdmat 20369, also a diagonal matrix). (Contributed by AV, 21-Dec-2019.)
(𝑅 ∈ Ring → ∅ ∈ (∅ ScMat 𝑅))

Theoremmat1scmat 20393 A 1-dimensional matrix over a ring is always a scalar matrix (and therefore, by scmatdmat 20369, 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 20238 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 20395 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 20151. See also the statements in [Lang] p. 508.

Syntaxcmvmul 20394 Syntax for the operator for the multiplication of a vector with a matrix.
class maVecMul

Definitiondf-mvmul 20395* 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 20396* 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 20397* 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 20398* 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 20399* 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 20400* 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)))))

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