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Theorem List for Metamath Proof Explorer - 20001-20100   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremlvecisfrlm 20001* Every vector space is isomorphic to a free module. (Contributed by AV, 7-Mar-2019.)
𝐹 = (Scalar‘𝑊)       (𝑊 ∈ LVec → ∃𝑘 𝑊𝑚 (𝐹 freeLMod 𝑘))

Theoremlmimco 20002 The composition of two isomorphisms of modules is an isomorphism of modules. (Contributed by AV, 10-Mar-2019.)
((𝐹 ∈ (𝑆 LMIso 𝑇) ∧ 𝐺 ∈ (𝑅 LMIso 𝑆)) → (𝐹𝐺) ∈ (𝑅 LMIso 𝑇))

Theoremlmictra 20003 Module isomorphism is transitive. (Contributed by AV, 10-Mar-2019.)
((𝑅𝑚 𝑆𝑆𝑚 𝑇) → 𝑅𝑚 𝑇)

Theoremuvcf1o 20004 In a nonzero ring, the mapping of the index set of a free module onto the unit vectors of the free module is a 1-1 onto function. (Contributed by AV, 10-Mar-2019.)
𝑈 = (𝑅 unitVec 𝐼)       ((𝑅 ∈ NzRing ∧ 𝐼𝑊) → 𝑈:𝐼1-1-onto→ran 𝑈)

Theoremuvcendim 20005 In a nonzero ring, the number of unit vectors of a free module corresponds to the dimension of the free module. (Contributed by AV, 10-Mar-2019.)
𝑈 = (𝑅 unitVec 𝐼)       ((𝑅 ∈ NzRing ∧ 𝐼𝑊) → 𝐼 ≈ ran 𝑈)

Theoremfrlmisfrlm 20006 A free module is isomorphic to a free module over the same (nonzero) ring, with the same cardinality. (Contributed by AV, 10-Mar-2019.)
((𝑅 ∈ NzRing ∧ 𝐼𝑌𝐼𝐽) → (𝑅 freeLMod 𝐼) ≃𝑚 (𝑅 freeLMod 𝐽))

Theoremfrlmiscvec 20007 Every free module is isomorphic to the free module of "column vectors" of the same dimension over the same (nonzero) ring. (Contributed by AV, 10-Mar-2019.)
((𝑅 ∈ NzRing ∧ 𝐼𝑌) → (𝑅 freeLMod 𝐼) ≃𝑚 (𝑅 freeLMod (𝐼 × {∅})))

11.2  Matrices

According to Wikipedia ("Matrix (mathemetics)", 02-Apr-2019, https://en.wikipedia.org/wiki/Matrix_(mathematics)) "A matrix is a rectangular array of numbers or other mathematical objects for which operations such as addition and multiplication are defined. Most commonly, a matrix over a field F is a rectangular array of scalars each of which is a member of F. The numbers, symbols or expressions in the matrix are called its entries or its elements. The horizontal and vertical lines of entries in a matrix are called rows and columns, respectively.", and in the definition of [Lang] p. 503 "By an m x n matrix in [a commutative ring] R one means a doubly indexed family of elements of R, (aij), (i= 1,..., m and j = 1,... n) ... We call the elements aij the coefficients or components of the matrix. A 1 x n matrix is called a row vector (of dimension, or size, n) and a m x 1 matrix is called a column vector (of dimension, or size, m). In general, we say that (m,n) is the size of the matrix, ...". In contrast to these definitions, we denote any free module over a (not necessarily commutative) ring (in the meaning of df-frlm 19910) with a Cartesian product as index set as "matrix". The two sets of the Cartesian product even need neither to be ordered or a range of (nonnegative/positive) integers nor finite. By this, the addition and scalar multiplication for matrices correspond to the addition (see frlmplusgval 19926) and scalar multiplication (see frlmvscafval 19928) for free modules. Actually, there isn't a definition for (arbitrary) matrices: Even the (general) matrix multiplication can be defined using functions from Cartesian products into a ring (which are elements of the base set of free modules), see df-mamu 20009. By this, a statement like "Then the set of m x n matrices in R is a module (i.e. an R-module)" as in [Lang] p. 504 follows immediatly from frlmlmod 19912.

However, for square matrices there is the definition df-mat 20033, defining the algebras of square matrices (of the same size over the same ring), extending the structure of the corresponding free module by the matrix multiplication as ring multiplication.

A "usual" matrix (aij), (i= 1,..., m and j = 1,... n) would be represented as element of (the base set of) (𝑅 freeLMod ((1...𝑚) × (1...𝑛))), and a square matrix (aij), (i= 1,..., n and j = 1,... n) would be represented as element of (the base set of) ((1...𝑛) Mat 𝑅).

Finally, it should be mentioned that our definitions of matrices include the zero-dimensional cases, which is excluded in the definition of many authors, e.g. in [Lang] p. 503. It is shown in mat0dimbas0 20091 that the empty set is the sole zero-dimensional matrix (also called "empty matrix", see Wikipedia https://en.wikipedia.org/wiki/Matrix_(mathematics)#Empty_matrices). The determinant is also defined for such an empty matrix, see mdet0pr 20217.

11.2.1  The matrix multiplication

This section is about the multiplication of m x n matrices.

Syntaxcmmul 20008 Syntax for the matrix multiplication operator.
class maMul

Definitiondf-mamu 20009* The operator which multiplies an m x n matrix with an n x p matrix, see also the definition in [Lang] p. 504. Note that it is not generally possible to recover the dimensions from the matrix, since all n x 0 and all 0 x n matrices are represented by the empty set. (Contributed by Stefan O'Rear, 4-Sep-2015.)
maMul = (𝑟 ∈ V, 𝑜 ∈ V ↦ (1st ‘(1st𝑜)) / 𝑚(2nd ‘(1st𝑜)) / 𝑛(2nd𝑜) / 𝑝(𝑥 ∈ ((Base‘𝑟) ↑𝑚 (𝑚 × 𝑛)), 𝑦 ∈ ((Base‘𝑟) ↑𝑚 (𝑛 × 𝑝)) ↦ (𝑖𝑚, 𝑘𝑝 ↦ (𝑟 Σg (𝑗𝑛 ↦ ((𝑖𝑥𝑗)(.r𝑟)(𝑗𝑦𝑘)))))))

Theoremmamufval 20010* Functional value of the matrix multiplication operator. (Contributed by Stefan O'Rear, 2-Sep-2015.)
𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑃⟩)    &   𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅𝑉)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑃 ∈ Fin)       (𝜑𝐹 = (𝑥 ∈ (𝐵𝑚 (𝑀 × 𝑁)), 𝑦 ∈ (𝐵𝑚 (𝑁 × 𝑃)) ↦ (𝑖𝑀, 𝑘𝑃 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑥𝑗) · (𝑗𝑦𝑘)))))))

Theoremmamuval 20011* Multiplication of two matrices. (Contributed by Stefan O'Rear, 2-Sep-2015.)
𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑃⟩)    &   𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅𝑉)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑃 ∈ Fin)    &   (𝜑𝑋 ∈ (𝐵𝑚 (𝑀 × 𝑁)))    &   (𝜑𝑌 ∈ (𝐵𝑚 (𝑁 × 𝑃)))       (𝜑 → (𝑋𝐹𝑌) = (𝑖𝑀, 𝑘𝑃 ↦ (𝑅 Σg (𝑗𝑁 ↦ ((𝑖𝑋𝑗) · (𝑗𝑌𝑘))))))

Theoremmamufv 20012* A cell in the multiplication of two matrices. (Contributed by Stefan O'Rear, 2-Sep-2015.)
𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑃⟩)    &   𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   (𝜑𝑅𝑉)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑃 ∈ Fin)    &   (𝜑𝑋 ∈ (𝐵𝑚 (𝑀 × 𝑁)))    &   (𝜑𝑌 ∈ (𝐵𝑚 (𝑁 × 𝑃)))    &   (𝜑𝐼𝑀)    &   (𝜑𝐾𝑃)       (𝜑 → (𝐼(𝑋𝐹𝑌)𝐾) = (𝑅 Σg (𝑗𝑁 ↦ ((𝐼𝑋𝑗) · (𝑗𝑌𝐾)))))

Theoremmamudm 20013 The domain of the matrix multiplication function. (Contributed by AV, 10-Feb-2019.)
𝐸 = (𝑅 freeLMod (𝑀 × 𝑁))    &   𝐵 = (Base‘𝐸)    &   𝐹 = (𝑅 freeLMod (𝑁 × 𝑃))    &   𝐶 = (Base‘𝐹)    &    × = (𝑅 maMul ⟨𝑀, 𝑁, 𝑃⟩)       ((𝑅𝑉 ∧ (𝑀 ∈ Fin ∧ 𝑁 ∈ Fin ∧ 𝑃 ∈ Fin)) → dom × = (𝐵 × 𝐶))

Theoremmamufacex 20014 Every solution of the equation 𝐴𝑋 = 𝐵 for matrices 𝐴 and 𝐵 is a matrix. (Contributed by AV, 10-Feb-2019.)
𝐸 = (𝑅 freeLMod (𝑀 × 𝑁))    &   𝐵 = (Base‘𝐸)    &   𝐹 = (𝑅 freeLMod (𝑁 × 𝑃))    &   𝐶 = (Base‘𝐹)    &    × = (𝑅 maMul ⟨𝑀, 𝑁, 𝑃⟩)    &   𝐺 = (𝑅 freeLMod (𝑀 × 𝑃))    &   𝐷 = (Base‘𝐺)       (((𝑀 ≠ ∅ ∧ 𝑃 ≠ ∅) ∧ (𝑅𝑉𝑌𝐷) ∧ (𝑀 ∈ Fin ∧ 𝑁 ∈ Fin ∧ 𝑃 ∈ Fin)) → ((𝑋 × 𝑍) = 𝑌𝑍𝐶))

Theoremmamures 20015 Rows in a matrix product are functions only of the corresponding rows in the left argument. (Contributed by SO, 9-Jul-2018.)
𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑃⟩)    &   𝐺 = (𝑅 maMul ⟨𝐼, 𝑁, 𝑃⟩)    &   𝐵 = (Base‘𝑅)    &   (𝜑𝑅𝑉)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑃 ∈ Fin)    &   (𝜑𝐼𝑀)    &   (𝜑𝑋 ∈ (𝐵𝑚 (𝑀 × 𝑁)))    &   (𝜑𝑌 ∈ (𝐵𝑚 (𝑁 × 𝑃)))       (𝜑 → ((𝑋𝐹𝑌) ↾ (𝐼 × 𝑃)) = ((𝑋 ↾ (𝐼 × 𝑁))𝐺𝑌))

Theoremmndvcl 20016 Tuple-wise additive closure in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝑀)    &    + = (+g𝑀)       ((𝑀 ∈ Mnd ∧ 𝑋 ∈ (𝐵𝑚 𝐼) ∧ 𝑌 ∈ (𝐵𝑚 𝐼)) → (𝑋𝑓 + 𝑌) ∈ (𝐵𝑚 𝐼))

Theoremmndvass 20017 Tuple-wise associativity in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝑀)    &    + = (+g𝑀)       ((𝑀 ∈ Mnd ∧ (𝑋 ∈ (𝐵𝑚 𝐼) ∧ 𝑌 ∈ (𝐵𝑚 𝐼) ∧ 𝑍 ∈ (𝐵𝑚 𝐼))) → ((𝑋𝑓 + 𝑌) ∘𝑓 + 𝑍) = (𝑋𝑓 + (𝑌𝑓 + 𝑍)))

Theoremmndvlid 20018 Tuple-wise left identity in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝑀)    &    + = (+g𝑀)    &    0 = (0g𝑀)       ((𝑀 ∈ Mnd ∧ 𝑋 ∈ (𝐵𝑚 𝐼)) → ((𝐼 × { 0 }) ∘𝑓 + 𝑋) = 𝑋)

Theoremmndvrid 20019 Tuple-wise right identity in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝑀)    &    + = (+g𝑀)    &    0 = (0g𝑀)       ((𝑀 ∈ Mnd ∧ 𝑋 ∈ (𝐵𝑚 𝐼)) → (𝑋𝑓 + (𝐼 × { 0 })) = 𝑋)

Theoremgrpvlinv 20020 Tuple-wise left inverse in groups. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝑁 = (invg𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵𝑚 𝐼)) → ((𝑁𝑋) ∘𝑓 + 𝑋) = (𝐼 × { 0 }))

Theoremgrpvrinv 20021 Tuple-wise right inverse in groups. (Contributed by Mario Carneiro, 22-Sep-2015.)
𝐵 = (Base‘𝐺)    &    + = (+g𝐺)    &   𝑁 = (invg𝐺)    &    0 = (0g𝐺)       ((𝐺 ∈ Grp ∧ 𝑋 ∈ (𝐵𝑚 𝐼)) → (𝑋𝑓 + (𝑁𝑋)) = (𝐼 × { 0 }))

Theoremmhmvlin 20022 Tuple extension of monoid homomorphisms. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝑀)    &    + = (+g𝑀)    &    = (+g𝑁)       ((𝐹 ∈ (𝑀 MndHom 𝑁) ∧ 𝑋 ∈ (𝐵𝑚 𝐼) ∧ 𝑌 ∈ (𝐵𝑚 𝐼)) → (𝐹 ∘ (𝑋𝑓 + 𝑌)) = ((𝐹𝑋) ∘𝑓 (𝐹𝑌)))

Theoremringvcl 20023 Tuple-wise multiplication closure in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝑅)    &    · = (.r𝑅)       ((𝑅 ∈ Ring ∧ 𝑋 ∈ (𝐵𝑚 𝐼) ∧ 𝑌 ∈ (𝐵𝑚 𝐼)) → (𝑋𝑓 · 𝑌) ∈ (𝐵𝑚 𝐼))

Theoremgsumcom3 20024* A commutative law for finitely supported iterated sums. (Contributed by Stefan O'Rear, 2-Nov-2015.)
𝐵 = (Base‘𝐺)    &    0 = (0g𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴𝑉)    &   (𝜑𝐶𝑊)    &   ((𝜑 ∧ (𝑗𝐴𝑘𝐶)) → 𝑋𝐵)    &   (𝜑𝑈 ∈ Fin)    &   ((𝜑 ∧ ((𝑗𝐴𝑘𝐶) ∧ ¬ 𝑗𝑈𝑘)) → 𝑋 = 0 )       (𝜑 → (𝐺 Σg (𝑗𝐴 ↦ (𝐺 Σg (𝑘𝐶𝑋)))) = (𝐺 Σg (𝑘𝐶 ↦ (𝐺 Σg (𝑗𝐴𝑋)))))

Theoremgsumcom3fi 20025* A commutative law for finite iterated sums. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝐺)    &   (𝜑𝐺 ∈ CMnd)    &   (𝜑𝐴 ∈ Fin)    &   (𝜑𝐶 ∈ Fin)    &   ((𝜑 ∧ (𝑗𝐴𝑘𝐶)) → 𝑋𝐵)       (𝜑 → (𝐺 Σg (𝑗𝐴 ↦ (𝐺 Σg (𝑘𝐶𝑋)))) = (𝐺 Σg (𝑘𝐶 ↦ (𝐺 Σg (𝑗𝐴𝑋)))))

Theoremmamucl 20026 Operation closure of matrix multiplication. (Contributed by Stefan O'Rear, 2-Sep-2015.) (Proof shortened by AV, 23-Jul-2019.)
𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑃⟩)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑃 ∈ Fin)    &   (𝜑𝑋 ∈ (𝐵𝑚 (𝑀 × 𝑁)))    &   (𝜑𝑌 ∈ (𝐵𝑚 (𝑁 × 𝑃)))       (𝜑 → (𝑋𝐹𝑌) ∈ (𝐵𝑚 (𝑀 × 𝑃)))

Theoremmamuass 20027 Matrix multiplication is associative, see also statement in [Lang] p. 505. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑂 ∈ Fin)    &   (𝜑𝑃 ∈ Fin)    &   (𝜑𝑋 ∈ (𝐵𝑚 (𝑀 × 𝑁)))    &   (𝜑𝑌 ∈ (𝐵𝑚 (𝑁 × 𝑂)))    &   (𝜑𝑍 ∈ (𝐵𝑚 (𝑂 × 𝑃)))    &   𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑂⟩)    &   𝐺 = (𝑅 maMul ⟨𝑀, 𝑂, 𝑃⟩)    &   𝐻 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑃⟩)    &   𝐼 = (𝑅 maMul ⟨𝑁, 𝑂, 𝑃⟩)       (𝜑 → ((𝑋𝐹𝑌)𝐺𝑍) = (𝑋𝐻(𝑌𝐼𝑍)))

Theoremmamudi 20028 Matrix multiplication distributes over addition on the left. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 23-Jul-2019.)
𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑂⟩)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑂 ∈ Fin)    &    + = (+g𝑅)    &   (𝜑𝑋 ∈ (𝐵𝑚 (𝑀 × 𝑁)))    &   (𝜑𝑌 ∈ (𝐵𝑚 (𝑀 × 𝑁)))    &   (𝜑𝑍 ∈ (𝐵𝑚 (𝑁 × 𝑂)))       (𝜑 → ((𝑋𝑓 + 𝑌)𝐹𝑍) = ((𝑋𝐹𝑍) ∘𝑓 + (𝑌𝐹𝑍)))

Theoremmamudir 20029 Matrix multiplication distributes over addition on the right. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 23-Jul-2019.)
𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑂⟩)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑂 ∈ Fin)    &    + = (+g𝑅)    &   (𝜑𝑋 ∈ (𝐵𝑚 (𝑀 × 𝑁)))    &   (𝜑𝑌 ∈ (𝐵𝑚 (𝑁 × 𝑂)))    &   (𝜑𝑍 ∈ (𝐵𝑚 (𝑁 × 𝑂)))       (𝜑 → (𝑋𝐹(𝑌𝑓 + 𝑍)) = ((𝑋𝐹𝑌) ∘𝑓 + (𝑋𝐹𝑍)))

Theoremmamuvs1 20030 Matrix multiplication distributes over scalar multiplication on the left. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &   𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑂⟩)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑂 ∈ Fin)    &    · = (.r𝑅)    &   (𝜑𝑋𝐵)    &   (𝜑𝑌 ∈ (𝐵𝑚 (𝑀 × 𝑁)))    &   (𝜑𝑍 ∈ (𝐵𝑚 (𝑁 × 𝑂)))       (𝜑 → ((((𝑀 × 𝑁) × {𝑋}) ∘𝑓 · 𝑌)𝐹𝑍) = (((𝑀 × 𝑂) × {𝑋}) ∘𝑓 · (𝑌𝐹𝑍)))

Theoremmamuvs2 20031 Matrix multiplication distributes over scalar multiplication on the left. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 22-Jul-2019.)
(𝜑𝑅 ∈ CRing)    &   𝐵 = (Base‘𝑅)    &    · = (.r𝑅)    &   𝐹 = (𝑅 maMul ⟨𝑀, 𝑁, 𝑂⟩)    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑂 ∈ Fin)    &   (𝜑𝑋 ∈ (𝐵𝑚 (𝑀 × 𝑁)))    &   (𝜑𝑌𝐵)    &   (𝜑𝑍 ∈ (𝐵𝑚 (𝑁 × 𝑂)))       (𝜑 → (𝑋𝐹(((𝑁 × 𝑂) × {𝑌}) ∘𝑓 · 𝑍)) = (((𝑀 × 𝑂) × {𝑌}) ∘𝑓 · (𝑋𝐹𝑍)))

11.2.2  Square matrices

In the following, the square matrix algebra is defined as extensible structure Mat. In this subsection, however, only square matrices and their basic properties are regarded. This includes showing that (𝑁 Mat 𝑅) is a left module, see matlmod 20054. That (𝑁 Mat 𝑅) is a ring and an associative algebra is shown in the next subsection, after theorems about the identity matrix are available. Nevertheless, (𝑁 Mat 𝑅) is called "matrix ring" or "matrix algebra" already in this subsection.

Syntaxcmat 20032 Syntax for the square matrix algebra.
class Mat

Definitiondf-mat 20033* Define the algebra of n x n matrices over a ring r. (Contributed by Stefan O'Rear, 31-Aug-2015.)
Mat = (𝑛 ∈ Fin, 𝑟 ∈ V ↦ ((𝑟 freeLMod (𝑛 × 𝑛)) sSet ⟨(.r‘ndx), (𝑟 maMul ⟨𝑛, 𝑛, 𝑛⟩)⟩))

Theoremmatbas0pc 20034 There is no matrix with a proper class either as dimension or as underlying ring. (Contributed by AV, 28-Dec-2018.)
(¬ (𝑁 ∈ V ∧ 𝑅 ∈ V) → (Base‘(𝑁 Mat 𝑅)) = ∅)

Theoremmatbas0 20035 There is no matrix for a not finite dimension or a proper class as the underlying ring. (Contributed by AV, 28-Dec-2018.)
(¬ (𝑁 ∈ Fin ∧ 𝑅 ∈ V) → (Base‘(𝑁 Mat 𝑅)) = ∅)

Theoremmatval 20036 Value of the matrix algebra. (Contributed by Stefan O'Rear, 4-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))    &    · = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝐴 = (𝐺 sSet ⟨(.r‘ndx), · ⟩))

Theoremmatrcl 20037 Reverse closure for the matrix algebra. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)       (𝑋𝐵 → (𝑁 ∈ Fin ∧ 𝑅 ∈ V))

Theoremmatbas 20038 The matrix ring has the same base set as its underlying group. (Contributed by Stefan O'Rear, 4-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (Base‘𝐺) = (Base‘𝐴))

Theoremmatplusg 20039 The matrix ring has the same addition as its underlying group. (Contributed by Stefan O'Rear, 4-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (+g𝐺) = (+g𝐴))

Theoremmatsca 20040 The matrix ring has the same scalars as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (Scalar‘𝐺) = (Scalar‘𝐴))

Theoremmatvsca 20041 The matrix ring has the same scalar multiplication as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → ( ·𝑠𝐺) = ( ·𝑠𝐴))

Theoremmat0 20042 The matrix ring has the same zero as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (0g𝐺) = (0g𝐴))

Theoremmatinvg 20043 The matrix ring has the same additive inverse as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (invg𝐺) = (invg𝐴))

Theoremmat0op 20044* Value of a zero matrix as operation. (Contributed by AV, 2-Dec-2018.)
𝐴 = (𝑁 Mat 𝑅)    &    0 = (0g𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → (0g𝐴) = (𝑖𝑁, 𝑗𝑁0 ))

Theoremmatsca2 20045 The scalars of the matrix ring are the underlying ring. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → 𝑅 = (Scalar‘𝐴))

Theoremmatbas2 20046 The base set of the matrix ring as a set exponential. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 16-Dec-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐾 = (Base‘𝑅)       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (𝐾𝑚 (𝑁 × 𝑁)) = (Base‘𝐴))

Theoremmatbas2i 20047 A matrix is a function. (Contributed by Stefan O'Rear, 11-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐾 = (Base‘𝑅)    &   𝐵 = (Base‘𝐴)       (𝑀𝐵𝑀 ∈ (𝐾𝑚 (𝑁 × 𝑁)))

Theoremmatbas2d 20048* The base set of the matrix ring as a mapping operation. (Contributed by Stefan O'Rear, 11-Jul-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐾 = (Base‘𝑅)    &   𝐵 = (Base‘𝐴)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝑅𝑉)    &   ((𝜑𝑥𝑁𝑦𝑁) → 𝐶𝐾)       (𝜑 → (𝑥𝑁, 𝑦𝑁𝐶) ∈ 𝐵)

Theoremeqmat 20049* Two square matrices of the same dimension are equal if they have the same entries. (Contributed by AV, 25-Sep-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)       ((𝑋𝐵𝑌𝐵) → (𝑋 = 𝑌 ↔ ∀𝑖𝑁𝑗𝑁 (𝑖𝑋𝑗) = (𝑖𝑌𝑗)))

Theoremmatecl 20050 Each entry (according to Wikipedia "Matrix (mathematics)", 30-Dec-2018, https://en.wikipedia.org/wiki/Matrix_(mathematics)#Definition (or element or component or coefficient or cell) of a matrix is an element of the underlying ring. (Contributed by AV, 16-Dec-2018.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐾 = (Base‘𝑅)       ((𝐼𝑁𝐽𝑁𝑀 ∈ (Base‘𝐴)) → (𝐼𝑀𝐽) ∈ 𝐾)

Theoremmatecld 20051 Each entry (according to Wikipedia "Matrix (mathematics)", 30-Dec-2018, https://en.wikipedia.org/wiki/Matrix_(mathematics)#Definition (or element or component or coefficient or cell) of a matrix is an element of the underlying ring, deduction form. (Contributed by AV, 27-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐾 = (Base‘𝑅)    &   𝐵 = (Base‘𝐴)    &   (𝜑𝐼𝑁)    &   (𝜑𝐽𝑁)    &   (𝜑𝑀𝐵)       (𝜑 → (𝐼𝑀𝐽) ∈ 𝐾)

Theoremmatplusg2 20052 Addition in the matrix ring is cell-wise. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    = (+g𝐴)    &    + = (+g𝑅)       ((𝑋𝐵𝑌𝐵) → (𝑋 𝑌) = (𝑋𝑓 + 𝑌))

Theoremmatvsca2 20053 Scalar multiplication in the matrix ring is cell-wise. (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐾 = (Base‘𝑅)    &    · = ( ·𝑠𝐴)    &    × = (.r𝑅)    &   𝐶 = (𝑁 × 𝑁)       ((𝑋𝐾𝑌𝐵) → (𝑋 · 𝑌) = ((𝐶 × {𝑋}) ∘𝑓 × 𝑌))

Theoremmatlmod 20054 The matrix ring is a linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ LMod)

Theoremmatgrp 20055 The matrix ring is a group. (Contributed by AV, 21-Dec-2019.)
𝐴 = (𝑁 Mat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Grp)

Theoremmatvscl 20056 Closure of the scalar multiplication in the matrix ring. (lmodvscl 18703 analog.) (Contributed by AV, 27-Nov-2019.)
𝐾 = (Base‘𝑅)    &   𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    · = ( ·𝑠𝐴)       (((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) ∧ (𝐶𝐾𝑋𝐵)) → (𝐶 · 𝑋) ∈ 𝐵)

Theoremmatsubg 20057 The matrix ring has the same addition as its underlying group. (Contributed by AV, 2-Aug-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐺 = (𝑅 freeLMod (𝑁 × 𝑁))       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → (-g𝐺) = (-g𝐴))

Theoremmatplusgcell 20058 Addition in the matrix ring is cell-wise. (Contributed by AV, 2-Aug-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    = (+g𝐴)    &    + = (+g𝑅)       (((𝑋𝐵𝑌𝐵) ∧ (𝐼𝑁𝐽𝑁)) → (𝐼(𝑋 𝑌)𝐽) = ((𝐼𝑋𝐽) + (𝐼𝑌𝐽)))

Theoremmatsubgcell 20059 Subtraction in the matrix ring is cell-wise. (Contributed by AV, 2-Aug-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑆 = (-g𝐴)    &    = (-g𝑅)       ((𝑅 ∈ Ring ∧ (𝑋𝐵𝑌𝐵) ∧ (𝐼𝑁𝐽𝑁)) → (𝐼(𝑋𝑆𝑌)𝐽) = ((𝐼𝑋𝐽) (𝐼𝑌𝐽)))

Theoremmatinvgcell 20060 Additive inversion in the matrix ring is cell-wise. (Contributed by AV, 17-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝑉 = (invg𝑅)    &   𝑊 = (invg𝐴)       ((𝑅 ∈ Ring ∧ 𝑋𝐵 ∧ (𝐼𝑁𝐽𝑁)) → (𝐼(𝑊𝑋)𝐽) = (𝑉‘(𝐼𝑋𝐽)))

Theoremmatvscacell 20061 Scalar multiplication in the matrix ring is cell-wise. (Contributed by AV, 7-Aug-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &   𝐾 = (Base‘𝑅)    &    · = ( ·𝑠𝐴)    &    × = (.r𝑅)       ((𝑅 ∈ Ring ∧ (𝑋𝐾𝑌𝐵) ∧ (𝐼𝑁𝐽𝑁)) → (𝐼(𝑋 · 𝑌)𝐽) = (𝑋 × (𝐼𝑌𝐽)))

Theoremmatgsum 20062* Finite commutative sums in a matrix algebra are taken componentwise. (Contributed by AV, 26-Sep-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    0 = (0g𝐴)    &   (𝜑𝑁 ∈ Fin)    &   (𝜑𝐽𝑊)    &   (𝜑𝑅 ∈ Ring)    &   ((𝜑𝑦𝐽) → (𝑖𝑁, 𝑗𝑁𝑈) ∈ 𝐵)    &   (𝜑 → (𝑦𝐽 ↦ (𝑖𝑁, 𝑗𝑁𝑈)) finSupp 0 )       (𝜑 → (𝐴 Σg (𝑦𝐽 ↦ (𝑖𝑁, 𝑗𝑁𝑈))) = (𝑖𝑁, 𝑗𝑁 ↦ (𝑅 Σg (𝑦𝐽𝑈))))

11.2.3  The matrix algebra

The main result of this subsection are the theorems showing that (𝑁 Mat 𝑅) is a ring (see matring 20068) and an associative algebra (see matassa 20069). Additionally, theorems for the identity matrix and transposed matrices are provided.

Theoremmatmulr 20063 Multiplication in the matrix algebra. (Contributed by Stefan O'Rear, 4-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)    &    · = (𝑅 maMul ⟨𝑁, 𝑁, 𝑁⟩)       ((𝑁 ∈ Fin ∧ 𝑅𝑉) → · = (.r𝐴))

Theoremmamumat1cl 20064* The identity matrix (as operation in maps-to notation) is a matrix. (Contributed by Stefan O'Rear, 2-Sep-2015.)
𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &    1 = (1r𝑅)    &    0 = (0g𝑅)    &   𝐼 = (𝑖𝑀, 𝑗𝑀 ↦ if(𝑖 = 𝑗, 1 , 0 ))    &   (𝜑𝑀 ∈ Fin)       (𝜑𝐼 ∈ (𝐵𝑚 (𝑀 × 𝑀)))

Theoremmat1comp 20065* The components of the identity matrix (as operation in maps-to notation). (Contributed by AV, 22-Jul-2019.)
𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &    1 = (1r𝑅)    &    0 = (0g𝑅)    &   𝐼 = (𝑖𝑀, 𝑗𝑀 ↦ if(𝑖 = 𝑗, 1 , 0 ))    &   (𝜑𝑀 ∈ Fin)       ((𝐴𝑀𝐽𝑀) → (𝐴𝐼𝐽) = if(𝐴 = 𝐽, 1 , 0 ))

Theoremmamulid 20066* The identity matrix (as operation in maps-to notation) is a left identity (for any matrix with the same number of rows). (Contributed by Stefan O'Rear, 3-Sep-2015.) (Proof shortened by AV, 22-Jul-2019.)
𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &    1 = (1r𝑅)    &    0 = (0g𝑅)    &   𝐼 = (𝑖𝑀, 𝑗𝑀 ↦ if(𝑖 = 𝑗, 1 , 0 ))    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   𝐹 = (𝑅 maMul ⟨𝑀, 𝑀, 𝑁⟩)    &   (𝜑𝑋 ∈ (𝐵𝑚 (𝑀 × 𝑁)))       (𝜑 → (𝐼𝐹𝑋) = 𝑋)

Theoremmamurid 20067* The identity matrix (as operation in maps-to notation) is a right identity (for any matrix with the same number of columns). (Contributed by Stefan O'Rear, 3-Sep-2015.) (Proof shortened by AV, 22-Jul-2019.)
𝐵 = (Base‘𝑅)    &   (𝜑𝑅 ∈ Ring)    &    1 = (1r𝑅)    &    0 = (0g𝑅)    &   𝐼 = (𝑖𝑀, 𝑗𝑀 ↦ if(𝑖 = 𝑗, 1 , 0 ))    &   (𝜑𝑀 ∈ Fin)    &   (𝜑𝑁 ∈ Fin)    &   𝐹 = (𝑅 maMul ⟨𝑁, 𝑀, 𝑀⟩)    &   (𝜑𝑋 ∈ (𝐵𝑚 (𝑁 × 𝑀)))       (𝜑 → (𝑋𝐹𝐼) = 𝑋)

Theoremmatring 20068 Existence of the matrix ring, see also the statement in [Lang] p. 504: "For a given integer n > 0 the set of square n x n matrices form a ring." (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ Ring) → 𝐴 ∈ Ring)

Theoremmatassa 20069 Existence of the matrix algebra, see also the statement in [Lang] p. 505:"Then Matn(R) is an algebra over R" . (Contributed by Stefan O'Rear, 5-Sep-2015.)
𝐴 = (𝑁 Mat 𝑅)       ((𝑁 ∈ Fin ∧ 𝑅 ∈ CRing) → 𝐴 ∈ AssAlg)

Theoremmatmulcell 20070* Multiplication in the matrix ring for a single cell of a matrix. (Contributed by AV, 17-Nov-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝐴)    &    · = (.r𝑅)    &    × = (.r𝐴)       ((𝑅 ∈ Ring ∧ (𝑋𝐵𝑌𝐵) ∧ (𝐼𝑁𝐽𝑁)) → (𝐼(𝑋 × 𝑌)𝐽) = (𝑅 Σg (𝑗𝑁 ↦ ((𝐼𝑋𝑗)(.r𝑅)(𝑗𝑌𝐽)))))

Theoremmpt2matmul 20071* Multiplication of two N x N matrices given in maps-to notation. (Contributed by AV, 29-Oct-2019.)
𝐴 = (𝑁 Mat 𝑅)    &   𝐵 = (Base‘𝑅)    &    × = (.r𝐴)    &    · = (.r𝑅)    &   (𝜑𝑅𝑉)    &   (𝜑𝑁 ∈ Fin)    &   𝑋 = (𝑖𝑁, 𝑗𝑁𝐶)    &   𝑌 = (𝑖𝑁, 𝑗𝑁𝐸)    &   ((𝜑𝑖𝑁𝑗𝑁) → 𝐶𝐵)    &   ((𝜑𝑖𝑁𝑗𝑁) → 𝐸𝐵)    &   ((𝜑 ∧ (𝑘 = 𝑖𝑚 = 𝑗)) → 𝐷 = 𝐶)    &   ((𝜑 ∧ (𝑚 = 𝑖𝑙 = 𝑗)) → 𝐹 = 𝐸)    &   ((𝜑𝑘𝑁𝑚𝑁) → 𝐷𝑈)    &   ((𝜑𝑚𝑁𝑙𝑁) → 𝐹𝑊)       (𝜑 → (𝑋 × 𝑌) = (𝑘𝑁, 𝑙𝑁 ↦ (𝑅 Σg (𝑚𝑁 ↦ (𝐷 · 𝐹)))))

Theoremmat1 20072* 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 20073 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 20074 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 20075* 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 20076 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 20077 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 20078 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 20079 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 20080 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 20081 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 20082 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 20083 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 20084 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 20085* 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 20086* 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 20087* 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 20088* 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 20089* 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 20090* 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 20091, 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 20095.

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 20112.

Theoremmat0dimbas0 20091 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 20092 The zero of the algebra of matrices with dimension 0. (Contributed by AV, 6-Aug-2019.)
𝐴 = (∅ Mat 𝑅)       (𝑅 ∈ Ring → (0g𝐴) = ∅)

Theoremmat0dimid 20093 The identity of the algebra of matrices with dimension 0. (Contributed by AV, 6-Aug-2019.)
𝐴 = (∅ Mat 𝑅)       (𝑅 ∈ Ring → (1r𝐴) = ∅)

Theoremmat0dimscm 20094 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 20095 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 20096* 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 20097 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 20098 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 20099 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 20100 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)⟩})

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