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Theorem List for Metamath Proof Explorer - 22101-22200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremevl1varpw 22101 Univariate polynomial evaluation maps the exponentiation of a variable to the exponentiation of the evaluated variable. Remark: in contrast to evl1gsumadd 22098, the proof is shorter using evls1varpw 22067 instead of proving it directly. (Contributed by AV, 15-Sep-2019.)
๐‘„ = (eval1โ€˜๐‘…)    &   ๐‘Š = (Poly1โ€˜๐‘…)    &   ๐บ = (mulGrpโ€˜๐‘Š)    &   ๐‘‹ = (var1โ€˜๐‘…)    &   ๐ต = (Baseโ€˜๐‘…)    &    โ†‘ = (.gโ€˜๐บ)    &   (๐œ‘ โ†’ ๐‘… โˆˆ CRing)    &   (๐œ‘ โ†’ ๐‘ โˆˆ โ„•0)    โ‡’   (๐œ‘ โ†’ (๐‘„โ€˜(๐‘ โ†‘ ๐‘‹)) = (๐‘(.gโ€˜(mulGrpโ€˜(๐‘… โ†‘s ๐ต)))(๐‘„โ€˜๐‘‹)))
 
Theoremevl1varpwval 22102 Value of a univariate polynomial evaluation mapping the exponentiation of a variable to the exponentiation of the evaluated variable. (Contributed by AV, 14-Sep-2019.)
๐‘„ = (eval1โ€˜๐‘…)    &   ๐‘Š = (Poly1โ€˜๐‘…)    &   ๐บ = (mulGrpโ€˜๐‘Š)    &   ๐‘‹ = (var1โ€˜๐‘…)    &   ๐ต = (Baseโ€˜๐‘…)    &    โ†‘ = (.gโ€˜๐บ)    &   (๐œ‘ โ†’ ๐‘… โˆˆ CRing)    &   (๐œ‘ โ†’ ๐‘ โˆˆ โ„•0)    &   (๐œ‘ โ†’ ๐ถ โˆˆ ๐ต)    &   ๐ป = (mulGrpโ€˜๐‘…)    &   ๐ธ = (.gโ€˜๐ป)    โ‡’   (๐œ‘ โ†’ ((๐‘„โ€˜(๐‘ โ†‘ ๐‘‹))โ€˜๐ถ) = (๐‘๐ธ๐ถ))
 
Theoremevl1scvarpw 22103 Univariate polynomial evaluation maps a multiple of an exponentiation of a variable to the multiple of an exponentiation of the evaluated variable. (Contributed by AV, 18-Sep-2019.)
๐‘„ = (eval1โ€˜๐‘…)    &   ๐‘Š = (Poly1โ€˜๐‘…)    &   ๐บ = (mulGrpโ€˜๐‘Š)    &   ๐‘‹ = (var1โ€˜๐‘…)    &   ๐ต = (Baseโ€˜๐‘…)    &    โ†‘ = (.gโ€˜๐บ)    &   (๐œ‘ โ†’ ๐‘… โˆˆ CRing)    &   (๐œ‘ โ†’ ๐‘ โˆˆ โ„•0)    &    ร— = ( ยท๐‘  โ€˜๐‘Š)    &   (๐œ‘ โ†’ ๐ด โˆˆ ๐ต)    &   ๐‘† = (๐‘… โ†‘s ๐ต)    &    โˆ™ = (.rโ€˜๐‘†)    &   ๐‘€ = (mulGrpโ€˜๐‘†)    &   ๐น = (.gโ€˜๐‘€)    โ‡’   (๐œ‘ โ†’ (๐‘„โ€˜(๐ด ร— (๐‘ โ†‘ ๐‘‹))) = ((๐ต ร— {๐ด}) โˆ™ (๐‘๐น(๐‘„โ€˜๐‘‹))))
 
Theoremevl1scvarpwval 22104 Value of a univariate polynomial evaluation mapping a multiple of an exponentiation of a variable to the multiple of the exponentiation of the evaluated variable. (Contributed by AV, 18-Sep-2019.)
๐‘„ = (eval1โ€˜๐‘…)    &   ๐‘Š = (Poly1โ€˜๐‘…)    &   ๐บ = (mulGrpโ€˜๐‘Š)    &   ๐‘‹ = (var1โ€˜๐‘…)    &   ๐ต = (Baseโ€˜๐‘…)    &    โ†‘ = (.gโ€˜๐บ)    &   (๐œ‘ โ†’ ๐‘… โˆˆ CRing)    &   (๐œ‘ โ†’ ๐‘ โˆˆ โ„•0)    &    ร— = ( ยท๐‘  โ€˜๐‘Š)    &   (๐œ‘ โ†’ ๐ด โˆˆ ๐ต)    &   (๐œ‘ โ†’ ๐ถ โˆˆ ๐ต)    &   ๐ป = (mulGrpโ€˜๐‘…)    &   ๐ธ = (.gโ€˜๐ป)    &    ยท = (.rโ€˜๐‘…)    โ‡’   (๐œ‘ โ†’ ((๐‘„โ€˜(๐ด ร— (๐‘ โ†‘ ๐‘‹)))โ€˜๐ถ) = (๐ด ยท (๐‘๐ธ๐ถ)))
 
Theoremevl1gsummon 22105* Value of a univariate polynomial evaluation mapping an additive group sum of a multiple of an exponentiation of a variable to a group sum of the multiple of the exponentiation of the evaluated variable. (Contributed by AV, 18-Sep-2019.)
๐‘„ = (eval1โ€˜๐‘…)    &   ๐พ = (Baseโ€˜๐‘…)    &   ๐‘Š = (Poly1โ€˜๐‘…)    &   ๐ต = (Baseโ€˜๐‘Š)    &   ๐‘‹ = (var1โ€˜๐‘…)    &   ๐ป = (mulGrpโ€˜๐‘…)    &   ๐ธ = (.gโ€˜๐ป)    &   ๐บ = (mulGrpโ€˜๐‘Š)    &    โ†‘ = (.gโ€˜๐บ)    &    ร— = ( ยท๐‘  โ€˜๐‘Š)    &    ยท = (.rโ€˜๐‘…)    &   (๐œ‘ โ†’ ๐‘… โˆˆ CRing)    &   (๐œ‘ โ†’ โˆ€๐‘ฅ โˆˆ ๐‘€ ๐ด โˆˆ ๐พ)    &   (๐œ‘ โ†’ ๐‘€ โІ โ„•0)    &   (๐œ‘ โ†’ ๐‘€ โˆˆ Fin)    &   (๐œ‘ โ†’ โˆ€๐‘ฅ โˆˆ ๐‘€ ๐‘ โˆˆ โ„•0)    &   (๐œ‘ โ†’ ๐ถ โˆˆ ๐พ)    โ‡’   (๐œ‘ โ†’ ((๐‘„โ€˜(๐‘Š ฮฃg (๐‘ฅ โˆˆ ๐‘€ โ†ฆ (๐ด ร— (๐‘ โ†‘ ๐‘‹)))))โ€˜๐ถ) = (๐‘… ฮฃg (๐‘ฅ โˆˆ ๐‘€ โ†ฆ (๐ด ยท (๐‘๐ธ๐ถ)))))
 
11.4  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 21522) 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 21539) and scalar multiplication (see frlmvscafval 21541) for free modules. Actually, there is no 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 22107. Thus, 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 immediately from frlmlmod 21524.

However, for square matrices there is Definition df-mat 22129, 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 an element of (the base set of) (๐‘… freeLMod ((1...๐‘š) ร— (1...๐‘›))) and a square matrix (aij), (i = 1,..., n and j = 1,... n) would be represented as an element of (the base set of) ((1...๐‘›) Mat ๐‘…).

Finally, it should be mentioned that our definitions of matrices include the zero-dimensional cases, which are excluded from the definitions of many authors, e.g., in [Lang] p. 503. It is shown in mat0dimbas0 22189 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). 22189 Its determinant is the ring unity, see mdet0fv0 22317.

 
11.4.1  The matrix multiplication

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

 
Syntaxcmmul 22106 Syntax for the matrix multiplication operator.
class maMul
 
Definitiondf-mamu 22107* 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 = (๐‘Ÿ โˆˆ V, ๐‘œ โˆˆ V โ†ฆ โฆ‹(1st โ€˜(1st โ€˜๐‘œ)) / ๐‘šโฆŒโฆ‹(2nd โ€˜(1st โ€˜๐‘œ)) / ๐‘›โฆŒโฆ‹(2nd โ€˜๐‘œ) / ๐‘โฆŒ(๐‘ฅ โˆˆ ((Baseโ€˜๐‘Ÿ) โ†‘m (๐‘š ร— ๐‘›)), ๐‘ฆ โˆˆ ((Baseโ€˜๐‘Ÿ) โ†‘m (๐‘› ร— ๐‘)) โ†ฆ (๐‘– โˆˆ ๐‘š, ๐‘˜ โˆˆ ๐‘ โ†ฆ (๐‘Ÿ ฮฃg (๐‘— โˆˆ ๐‘› โ†ฆ ((๐‘–๐‘ฅ๐‘—)(.rโ€˜๐‘Ÿ)(๐‘—๐‘ฆ๐‘˜)))))))
 
Theoremmamufval 22108* Functional value of the matrix multiplication operator. (Contributed by Stefan O'Rear, 2-Sep-2015.)
๐น = (๐‘… maMul โŸจ๐‘€, ๐‘, ๐‘ƒโŸฉ)    &   ๐ต = (Baseโ€˜๐‘…)    &    ยท = (.rโ€˜๐‘…)    &   (๐œ‘ โ†’ ๐‘… โˆˆ ๐‘‰)    &   (๐œ‘ โ†’ ๐‘€ โˆˆ Fin)    &   (๐œ‘ โ†’ ๐‘ โˆˆ Fin)    &   (๐œ‘ โ†’ ๐‘ƒ โˆˆ Fin)    โ‡’   (๐œ‘ โ†’ ๐น = (๐‘ฅ โˆˆ (๐ต โ†‘m (๐‘€ ร— ๐‘)), ๐‘ฆ โˆˆ (๐ต โ†‘m (๐‘ ร— ๐‘ƒ)) โ†ฆ (๐‘– โˆˆ ๐‘€, ๐‘˜ โˆˆ ๐‘ƒ โ†ฆ (๐‘… ฮฃg (๐‘— โˆˆ ๐‘ โ†ฆ ((๐‘–๐‘ฅ๐‘—) ยท (๐‘—๐‘ฆ๐‘˜)))))))
 
Theoremmamuval 22109* Multiplication of two matrices. (Contributed by Stefan O'Rear, 2-Sep-2015.)
๐น = (๐‘… maMul โŸจ๐‘€, ๐‘, ๐‘ƒโŸฉ)    &   ๐ต = (Baseโ€˜๐‘…)    &    ยท = (.rโ€˜๐‘…)    &   (๐œ‘ โ†’ ๐‘… โˆˆ ๐‘‰)    &   (๐œ‘ โ†’ ๐‘€ โˆˆ Fin)    &   (๐œ‘ โ†’ ๐‘ โˆˆ Fin)    &   (๐œ‘ โ†’ ๐‘ƒ โˆˆ Fin)    &   (๐œ‘ โ†’ ๐‘‹ โˆˆ (๐ต โ†‘m (๐‘€ ร— ๐‘)))    &   (๐œ‘ โ†’ ๐‘Œ โˆˆ (๐ต โ†‘m (๐‘ ร— ๐‘ƒ)))    โ‡’   (๐œ‘ โ†’ (๐‘‹๐น๐‘Œ) = (๐‘– โˆˆ ๐‘€, ๐‘˜ โˆˆ ๐‘ƒ โ†ฆ (๐‘… ฮฃg (๐‘— โˆˆ ๐‘ โ†ฆ ((๐‘–๐‘‹๐‘—) ยท (๐‘—๐‘Œ๐‘˜))))))
 
Theoremmamufv 22110* A cell in the multiplication of two matrices. (Contributed by Stefan O'Rear, 2-Sep-2015.)
๐น = (๐‘… maMul โŸจ๐‘€, ๐‘, ๐‘ƒโŸฉ)    &   ๐ต = (Baseโ€˜๐‘…)    &    ยท = (.rโ€˜๐‘…)    &   (๐œ‘ โ†’ ๐‘… โˆˆ ๐‘‰)    &   (๐œ‘ โ†’ ๐‘€ โˆˆ Fin)    &   (๐œ‘ โ†’ ๐‘ โˆˆ Fin)    &   (๐œ‘ โ†’ ๐‘ƒ โˆˆ Fin)    &   (๐œ‘ โ†’ ๐‘‹ โˆˆ (๐ต โ†‘m (๐‘€ ร— ๐‘)))    &   (๐œ‘ โ†’ ๐‘Œ โˆˆ (๐ต โ†‘m (๐‘ ร— ๐‘ƒ)))    &   (๐œ‘ โ†’ ๐ผ โˆˆ ๐‘€)    &   (๐œ‘ โ†’ ๐พ โˆˆ ๐‘ƒ)    โ‡’   (๐œ‘ โ†’ (๐ผ(๐‘‹๐น๐‘Œ)๐พ) = (๐‘… ฮฃg (๐‘— โˆˆ ๐‘ โ†ฆ ((๐ผ๐‘‹๐‘—) ยท (๐‘—๐‘Œ๐พ)))))
 
Theoremmamudm 22111 The domain of the matrix multiplication function. (Contributed by AV, 10-Feb-2019.)
๐ธ = (๐‘… freeLMod (๐‘€ ร— ๐‘))    &   ๐ต = (Baseโ€˜๐ธ)    &   ๐น = (๐‘… freeLMod (๐‘ ร— ๐‘ƒ))    &   ๐ถ = (Baseโ€˜๐น)    &    ร— = (๐‘… maMul โŸจ๐‘€, ๐‘, ๐‘ƒโŸฉ)    โ‡’   ((๐‘… โˆˆ ๐‘‰ โˆง (๐‘€ โˆˆ Fin โˆง ๐‘ โˆˆ Fin โˆง ๐‘ƒ โˆˆ Fin)) โ†’ dom ร— = (๐ต ร— ๐ถ))
 
Theoremmamufacex 22112 Every solution of the equation ๐ดโˆ—๐‘‹ = ๐ต for matrices ๐ด and ๐ต is a matrix. (Contributed by AV, 10-Feb-2019.)
๐ธ = (๐‘… freeLMod (๐‘€ ร— ๐‘))    &   ๐ต = (Baseโ€˜๐ธ)    &   ๐น = (๐‘… freeLMod (๐‘ ร— ๐‘ƒ))    &   ๐ถ = (Baseโ€˜๐น)    &    ร— = (๐‘… maMul โŸจ๐‘€, ๐‘, ๐‘ƒโŸฉ)    &   ๐บ = (๐‘… freeLMod (๐‘€ ร— ๐‘ƒ))    &   ๐ท = (Baseโ€˜๐บ)    โ‡’   (((๐‘€ โ‰  โˆ… โˆง ๐‘ƒ โ‰  โˆ…) โˆง (๐‘… โˆˆ ๐‘‰ โˆง ๐‘Œ โˆˆ ๐ท) โˆง (๐‘€ โˆˆ Fin โˆง ๐‘ โˆˆ Fin โˆง ๐‘ƒ โˆˆ Fin)) โ†’ ((๐‘‹ ร— ๐‘) = ๐‘Œ โ†’ ๐‘ โˆˆ ๐ถ))
 
Theoremmamures 22113 Rows in a matrix product are functions only of the corresponding rows in the left argument. (Contributed by SO, 9-Jul-2018.)
๐น = (๐‘… maMul โŸจ๐‘€, ๐‘, ๐‘ƒโŸฉ)    &   ๐บ = (๐‘… maMul โŸจ๐ผ, ๐‘, ๐‘ƒโŸฉ)    &   ๐ต = (Baseโ€˜๐‘…)    &   (๐œ‘ โ†’ ๐‘… โˆˆ ๐‘‰)    &   (๐œ‘ โ†’ ๐‘€ โˆˆ Fin)    &   (๐œ‘ โ†’ ๐‘ โˆˆ Fin)    &   (๐œ‘ โ†’ ๐‘ƒ โˆˆ Fin)    &   (๐œ‘ โ†’ ๐ผ โІ ๐‘€)    &   (๐œ‘ โ†’ ๐‘‹ โˆˆ (๐ต โ†‘m (๐‘€ ร— ๐‘)))    &   (๐œ‘ โ†’ ๐‘Œ โˆˆ (๐ต โ†‘m (๐‘ ร— ๐‘ƒ)))    โ‡’   (๐œ‘ โ†’ ((๐‘‹๐น๐‘Œ) โ†พ (๐ผ ร— ๐‘ƒ)) = ((๐‘‹ โ†พ (๐ผ ร— ๐‘))๐บ๐‘Œ))
 
Theoremmndvcl 22114 Tuple-wise additive closure in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)
๐ต = (Baseโ€˜๐‘€)    &    + = (+gโ€˜๐‘€)    โ‡’   ((๐‘€ โˆˆ Mnd โˆง ๐‘‹ โˆˆ (๐ต โ†‘m ๐ผ) โˆง ๐‘Œ โˆˆ (๐ต โ†‘m ๐ผ)) โ†’ (๐‘‹ โˆ˜f + ๐‘Œ) โˆˆ (๐ต โ†‘m ๐ผ))
 
Theoremmndvass 22115 Tuple-wise associativity in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)
๐ต = (Baseโ€˜๐‘€)    &    + = (+gโ€˜๐‘€)    โ‡’   ((๐‘€ โˆˆ Mnd โˆง (๐‘‹ โˆˆ (๐ต โ†‘m ๐ผ) โˆง ๐‘Œ โˆˆ (๐ต โ†‘m ๐ผ) โˆง ๐‘ โˆˆ (๐ต โ†‘m ๐ผ))) โ†’ ((๐‘‹ โˆ˜f + ๐‘Œ) โˆ˜f + ๐‘) = (๐‘‹ โˆ˜f + (๐‘Œ โˆ˜f + ๐‘)))
 
Theoremmndvlid 22116 Tuple-wise left identity in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)
๐ต = (Baseโ€˜๐‘€)    &    + = (+gโ€˜๐‘€)    &    0 = (0gโ€˜๐‘€)    โ‡’   ((๐‘€ โˆˆ Mnd โˆง ๐‘‹ โˆˆ (๐ต โ†‘m ๐ผ)) โ†’ ((๐ผ ร— { 0 }) โˆ˜f + ๐‘‹) = ๐‘‹)
 
Theoremmndvrid 22117 Tuple-wise right identity in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)
๐ต = (Baseโ€˜๐‘€)    &    + = (+gโ€˜๐‘€)    &    0 = (0gโ€˜๐‘€)    โ‡’   ((๐‘€ โˆˆ Mnd โˆง ๐‘‹ โˆˆ (๐ต โ†‘m ๐ผ)) โ†’ (๐‘‹ โˆ˜f + (๐ผ ร— { 0 })) = ๐‘‹)
 
Theoremgrpvlinv 22118 Tuple-wise left inverse in groups. (Contributed by Stefan O'Rear, 5-Sep-2015.)
๐ต = (Baseโ€˜๐บ)    &    + = (+gโ€˜๐บ)    &   ๐‘ = (invgโ€˜๐บ)    &    0 = (0gโ€˜๐บ)    โ‡’   ((๐บ โˆˆ Grp โˆง ๐‘‹ โˆˆ (๐ต โ†‘m ๐ผ)) โ†’ ((๐‘ โˆ˜ ๐‘‹) โˆ˜f + ๐‘‹) = (๐ผ ร— { 0 }))
 
Theoremgrpvrinv 22119 Tuple-wise right inverse in groups. (Contributed by Mario Carneiro, 22-Sep-2015.)
๐ต = (Baseโ€˜๐บ)    &    + = (+gโ€˜๐บ)    &   ๐‘ = (invgโ€˜๐บ)    &    0 = (0gโ€˜๐บ)    โ‡’   ((๐บ โˆˆ Grp โˆง ๐‘‹ โˆˆ (๐ต โ†‘m ๐ผ)) โ†’ (๐‘‹ โˆ˜f + (๐‘ โˆ˜ ๐‘‹)) = (๐ผ ร— { 0 }))
 
Theoremmhmvlin 22120 Tuple extension of monoid homomorphisms. (Contributed by Stefan O'Rear, 5-Sep-2015.)
๐ต = (Baseโ€˜๐‘€)    &    + = (+gโ€˜๐‘€)    &    โจฃ = (+gโ€˜๐‘)    โ‡’   ((๐น โˆˆ (๐‘€ MndHom ๐‘) โˆง ๐‘‹ โˆˆ (๐ต โ†‘m ๐ผ) โˆง ๐‘Œ โˆˆ (๐ต โ†‘m ๐ผ)) โ†’ (๐น โˆ˜ (๐‘‹ โˆ˜f + ๐‘Œ)) = ((๐น โˆ˜ ๐‘‹) โˆ˜f โจฃ (๐น โˆ˜ ๐‘Œ)))
 
Theoremringvcl 22121 Tuple-wise multiplication closure in monoids. (Contributed by Stefan O'Rear, 5-Sep-2015.)
๐ต = (Baseโ€˜๐‘…)    &    ยท = (.rโ€˜๐‘…)    โ‡’   ((๐‘… โˆˆ Ring โˆง ๐‘‹ โˆˆ (๐ต โ†‘m ๐ผ) โˆง ๐‘Œ โˆˆ (๐ต โ†‘m ๐ผ)) โ†’ (๐‘‹ โˆ˜f ยท ๐‘Œ) โˆˆ (๐ต โ†‘m ๐ผ))
 
Theoremmamucl 22122 Operation closure of matrix multiplication. (Contributed by Stefan O'Rear, 2-Sep-2015.) (Proof shortened by AV, 23-Jul-2019.)
๐ต = (Baseโ€˜๐‘…)    &   (๐œ‘ โ†’ ๐‘… โˆˆ Ring)    &   ๐น = (๐‘… maMul โŸจ๐‘€, ๐‘, ๐‘ƒโŸฉ)    &   (๐œ‘ โ†’ ๐‘€ โˆˆ Fin)    &   (๐œ‘ โ†’ ๐‘ โˆˆ Fin)    &   (๐œ‘ โ†’ ๐‘ƒ โˆˆ Fin)    &   (๐œ‘ โ†’ ๐‘‹ โˆˆ (๐ต โ†‘m (๐‘€ ร— ๐‘)))    &   (๐œ‘ โ†’ ๐‘Œ โˆˆ (๐ต โ†‘m (๐‘ ร— ๐‘ƒ)))    โ‡’   (๐œ‘ โ†’ (๐‘‹๐น๐‘Œ) โˆˆ (๐ต โ†‘m (๐‘€ ร— ๐‘ƒ)))
 
Theoremmamuass 22123 Matrix multiplication is associative, see also statement in [Lang] p. 505. (Contributed by Stefan O'Rear, 5-Sep-2015.)
๐ต = (Baseโ€˜๐‘…)    &   (๐œ‘ โ†’ ๐‘… โˆˆ Ring)    &   (๐œ‘ โ†’ ๐‘€ โˆˆ Fin)    &   (๐œ‘ โ†’ ๐‘ โˆˆ Fin)    &   (๐œ‘ โ†’ ๐‘‚ โˆˆ Fin)    &   (๐œ‘ โ†’ ๐‘ƒ โˆˆ Fin)    &   (๐œ‘ โ†’ ๐‘‹ โˆˆ (๐ต โ†‘m (๐‘€ ร— ๐‘)))    &   (๐œ‘ โ†’ ๐‘Œ โˆˆ (๐ต โ†‘m (๐‘ ร— ๐‘‚)))    &   (๐œ‘ โ†’ ๐‘ โˆˆ (๐ต โ†‘m (๐‘‚ ร— ๐‘ƒ)))    &   ๐น = (๐‘… maMul โŸจ๐‘€, ๐‘, ๐‘‚โŸฉ)    &   ๐บ = (๐‘… maMul โŸจ๐‘€, ๐‘‚, ๐‘ƒโŸฉ)    &   ๐ป = (๐‘… maMul โŸจ๐‘€, ๐‘, ๐‘ƒโŸฉ)    &   ๐ผ = (๐‘… maMul โŸจ๐‘, ๐‘‚, ๐‘ƒโŸฉ)    โ‡’   (๐œ‘ โ†’ ((๐‘‹๐น๐‘Œ)๐บ๐‘) = (๐‘‹๐ป(๐‘Œ๐ผ๐‘)))
 
Theoremmamudi 22124 Matrix multiplication distributes over addition on the left. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 23-Jul-2019.)
๐ต = (Baseโ€˜๐‘…)    &   (๐œ‘ โ†’ ๐‘… โˆˆ Ring)    &   ๐น = (๐‘… maMul โŸจ๐‘€, ๐‘, ๐‘‚โŸฉ)    &   (๐œ‘ โ†’ ๐‘€ โˆˆ Fin)    &   (๐œ‘ โ†’ ๐‘ โˆˆ Fin)    &   (๐œ‘ โ†’ ๐‘‚ โˆˆ Fin)    &    + = (+gโ€˜๐‘…)    &   (๐œ‘ โ†’ ๐‘‹ โˆˆ (๐ต โ†‘m (๐‘€ ร— ๐‘)))    &   (๐œ‘ โ†’ ๐‘Œ โˆˆ (๐ต โ†‘m (๐‘€ ร— ๐‘)))    &   (๐œ‘ โ†’ ๐‘ โˆˆ (๐ต โ†‘m (๐‘ ร— ๐‘‚)))    โ‡’   (๐œ‘ โ†’ ((๐‘‹ โˆ˜f + ๐‘Œ)๐น๐‘) = ((๐‘‹๐น๐‘) โˆ˜f + (๐‘Œ๐น๐‘)))
 
Theoremmamudir 22125 Matrix multiplication distributes over addition on the right. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 23-Jul-2019.)
๐ต = (Baseโ€˜๐‘…)    &   (๐œ‘ โ†’ ๐‘… โˆˆ Ring)    &   ๐น = (๐‘… maMul โŸจ๐‘€, ๐‘, ๐‘‚โŸฉ)    &   (๐œ‘ โ†’ ๐‘€ โˆˆ Fin)    &   (๐œ‘ โ†’ ๐‘ โˆˆ Fin)    &   (๐œ‘ โ†’ ๐‘‚ โˆˆ Fin)    &    + = (+gโ€˜๐‘…)    &   (๐œ‘ โ†’ ๐‘‹ โˆˆ (๐ต โ†‘m (๐‘€ ร— ๐‘)))    &   (๐œ‘ โ†’ ๐‘Œ โˆˆ (๐ต โ†‘m (๐‘ ร— ๐‘‚)))    &   (๐œ‘ โ†’ ๐‘ โˆˆ (๐ต โ†‘m (๐‘ ร— ๐‘‚)))    โ‡’   (๐œ‘ โ†’ (๐‘‹๐น(๐‘Œ โˆ˜f + ๐‘)) = ((๐‘‹๐น๐‘Œ) โˆ˜f + (๐‘‹๐น๐‘)))
 
Theoremmamuvs1 22126 Matrix multiplication distributes over scalar multiplication on the left. (Contributed by Stefan O'Rear, 5-Sep-2015.)
๐ต = (Baseโ€˜๐‘…)    &   (๐œ‘ โ†’ ๐‘… โˆˆ Ring)    &   ๐น = (๐‘… maMul โŸจ๐‘€, ๐‘, ๐‘‚โŸฉ)    &   (๐œ‘ โ†’ ๐‘€ โˆˆ Fin)    &   (๐œ‘ โ†’ ๐‘ โˆˆ Fin)    &   (๐œ‘ โ†’ ๐‘‚ โˆˆ Fin)    &    ยท = (.rโ€˜๐‘…)    &   (๐œ‘ โ†’ ๐‘‹ โˆˆ ๐ต)    &   (๐œ‘ โ†’ ๐‘Œ โˆˆ (๐ต โ†‘m (๐‘€ ร— ๐‘)))    &   (๐œ‘ โ†’ ๐‘ โˆˆ (๐ต โ†‘m (๐‘ ร— ๐‘‚)))    โ‡’   (๐œ‘ โ†’ ((((๐‘€ ร— ๐‘) ร— {๐‘‹}) โˆ˜f ยท ๐‘Œ)๐น๐‘) = (((๐‘€ ร— ๐‘‚) ร— {๐‘‹}) โˆ˜f ยท (๐‘Œ๐น๐‘)))
 
Theoremmamuvs2 22127 Matrix multiplication distributes over scalar multiplication on the left. (Contributed by Stefan O'Rear, 5-Sep-2015.) (Proof shortened by AV, 22-Jul-2019.)
(๐œ‘ โ†’ ๐‘… โˆˆ CRing)    &   ๐ต = (Baseโ€˜๐‘…)    &    ยท = (.rโ€˜๐‘…)    &   ๐น = (๐‘… maMul โŸจ๐‘€, ๐‘, ๐‘‚โŸฉ)    &   (๐œ‘ โ†’ ๐‘€ โˆˆ Fin)    &   (๐œ‘ โ†’ ๐‘ โˆˆ Fin)    &   (๐œ‘ โ†’ ๐‘‚ โˆˆ Fin)    &   (๐œ‘ โ†’ ๐‘‹ โˆˆ (๐ต โ†‘m (๐‘€ ร— ๐‘)))    &   (๐œ‘ โ†’ ๐‘Œ โˆˆ ๐ต)    &   (๐œ‘ โ†’ ๐‘ โˆˆ (๐ต โ†‘m (๐‘ ร— ๐‘‚)))    โ‡’   (๐œ‘ โ†’ (๐‘‹๐น(((๐‘ ร— ๐‘‚) ร— {๐‘Œ}) โˆ˜f ยท ๐‘)) = (((๐‘€ ร— ๐‘‚) ร— {๐‘Œ}) โˆ˜f ยท (๐‘‹๐น๐‘)))
 
11.4.2  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 (๐‘ Mat ๐‘…) is a left module, see matlmod 22152. That (๐‘ Mat ๐‘…) is a ring and an associative algebra is shown in the next subsection, after theorems about the identity matrix are available. Nevertheless, (๐‘ Mat ๐‘…) is called "matrix ring" or "matrix algebra" already in this subsection.

 
Syntaxcmat 22128 Syntax for the square matrix algebra.
class Mat
 
Definitiondf-mat 22129* Define the algebra of n x n matrices over a ring r. (Contributed by Stefan O'Rear, 31-Aug-2015.)
Mat = (๐‘› โˆˆ Fin, ๐‘Ÿ โˆˆ V โ†ฆ ((๐‘Ÿ freeLMod (๐‘› ร— ๐‘›)) sSet โŸจ(.rโ€˜ndx), (๐‘Ÿ maMul โŸจ๐‘›, ๐‘›, ๐‘›โŸฉ)โŸฉ))
 
Theoremmatbas0pc 22130 There is no matrix with a proper class either as dimension or as underlying ring. (Contributed by AV, 28-Dec-2018.)
(ยฌ (๐‘ โˆˆ V โˆง ๐‘… โˆˆ V) โ†’ (Baseโ€˜(๐‘ Mat ๐‘…)) = โˆ…)
 
Theoremmatbas0 22131 There is no matrix for a not finite dimension or a proper class as the underlying ring. (Contributed by AV, 28-Dec-2018.)
(ยฌ (๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ V) โ†’ (Baseโ€˜(๐‘ Mat ๐‘…)) = โˆ…)
 
Theoremmatval 22132 Value of the matrix algebra. (Contributed by Stefan O'Rear, 4-Sep-2015.)
๐ด = (๐‘ Mat ๐‘…)    &   ๐บ = (๐‘… freeLMod (๐‘ ร— ๐‘))    &    ยท = (๐‘… maMul โŸจ๐‘, ๐‘, ๐‘โŸฉ)    โ‡’   ((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ ๐‘‰) โ†’ ๐ด = (๐บ sSet โŸจ(.rโ€˜ndx), ยท โŸฉ))
 
Theoremmatrcl 22133 Reverse closure for the matrix algebra. (Contributed by Stefan O'Rear, 5-Sep-2015.)
๐ด = (๐‘ Mat ๐‘…)    &   ๐ต = (Baseโ€˜๐ด)    โ‡’   (๐‘‹ โˆˆ ๐ต โ†’ (๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ V))
 
Theoremmatbas 22134 The matrix ring has the same base set as its underlying group. (Contributed by Stefan O'Rear, 4-Sep-2015.)
๐ด = (๐‘ Mat ๐‘…)    &   ๐บ = (๐‘… freeLMod (๐‘ ร— ๐‘))    โ‡’   ((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ ๐‘‰) โ†’ (Baseโ€˜๐บ) = (Baseโ€˜๐ด))
 
Theoremmatplusg 22135 The matrix ring has the same addition as its underlying group. (Contributed by Stefan O'Rear, 4-Sep-2015.)
๐ด = (๐‘ Mat ๐‘…)    &   ๐บ = (๐‘… freeLMod (๐‘ ร— ๐‘))    โ‡’   ((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ ๐‘‰) โ†’ (+gโ€˜๐บ) = (+gโ€˜๐ด))
 
Theoremmatsca 22136 The matrix ring has the same scalars as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.) (Proof shortened by AV, 12-Nov-2024.)
๐ด = (๐‘ Mat ๐‘…)    &   ๐บ = (๐‘… freeLMod (๐‘ ร— ๐‘))    โ‡’   ((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ ๐‘‰) โ†’ (Scalarโ€˜๐บ) = (Scalarโ€˜๐ด))
 
TheoremmatscaOLD 22137 Obsolete proof of matsca 22136 as of 12-Nov-2024. The matrix ring has the same scalars as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
๐ด = (๐‘ Mat ๐‘…)    &   ๐บ = (๐‘… freeLMod (๐‘ ร— ๐‘))    โ‡’   ((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ ๐‘‰) โ†’ (Scalarโ€˜๐บ) = (Scalarโ€˜๐ด))
 
Theoremmatvsca 22138 The matrix ring has the same scalar multiplication as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.) (Proof shortened by AV, 12-Nov-2024.)
๐ด = (๐‘ Mat ๐‘…)    &   ๐บ = (๐‘… freeLMod (๐‘ ร— ๐‘))    โ‡’   ((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ ๐‘‰) โ†’ ( ยท๐‘  โ€˜๐บ) = ( ยท๐‘  โ€˜๐ด))
 
TheoremmatvscaOLD 22139 Obsolete proof of matvsca 22138 as of 12-Nov-2024. The matrix ring has the same scalar multiplication as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
๐ด = (๐‘ Mat ๐‘…)    &   ๐บ = (๐‘… freeLMod (๐‘ ร— ๐‘))    โ‡’   ((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ ๐‘‰) โ†’ ( ยท๐‘  โ€˜๐บ) = ( ยท๐‘  โ€˜๐ด))
 
Theoremmat0 22140 The matrix ring has the same zero as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.)
๐ด = (๐‘ Mat ๐‘…)    &   ๐บ = (๐‘… freeLMod (๐‘ ร— ๐‘))    โ‡’   ((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ ๐‘‰) โ†’ (0gโ€˜๐บ) = (0gโ€˜๐ด))
 
Theoremmatinvg 22141 The matrix ring has the same additive inverse as its underlying linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.)
๐ด = (๐‘ Mat ๐‘…)    &   ๐บ = (๐‘… freeLMod (๐‘ ร— ๐‘))    โ‡’   ((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ ๐‘‰) โ†’ (invgโ€˜๐บ) = (invgโ€˜๐ด))
 
Theoremmat0op 22142* Value of a zero matrix as operation. (Contributed by AV, 2-Dec-2018.)
๐ด = (๐‘ Mat ๐‘…)    &    0 = (0gโ€˜๐‘…)    โ‡’   ((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ Ring) โ†’ (0gโ€˜๐ด) = (๐‘– โˆˆ ๐‘, ๐‘— โˆˆ ๐‘ โ†ฆ 0 ))
 
Theoremmatsca2 22143 The scalars of the matrix ring are the underlying ring. (Contributed by Stefan O'Rear, 5-Sep-2015.)
๐ด = (๐‘ Mat ๐‘…)    โ‡’   ((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ ๐‘‰) โ†’ ๐‘… = (Scalarโ€˜๐ด))
 
Theoremmatbas2 22144 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.)
๐ด = (๐‘ Mat ๐‘…)    &   ๐พ = (Baseโ€˜๐‘…)    โ‡’   ((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ ๐‘‰) โ†’ (๐พ โ†‘m (๐‘ ร— ๐‘)) = (Baseโ€˜๐ด))
 
Theoremmatbas2i 22145 A matrix is a function. (Contributed by Stefan O'Rear, 11-Sep-2015.)
๐ด = (๐‘ Mat ๐‘…)    &   ๐พ = (Baseโ€˜๐‘…)    &   ๐ต = (Baseโ€˜๐ด)    โ‡’   (๐‘€ โˆˆ ๐ต โ†’ ๐‘€ โˆˆ (๐พ โ†‘m (๐‘ ร— ๐‘)))
 
Theoremmatbas2d 22146* The base set of the matrix ring as a mapping operation. (Contributed by Stefan O'Rear, 11-Jul-2018.)
๐ด = (๐‘ Mat ๐‘…)    &   ๐พ = (Baseโ€˜๐‘…)    &   ๐ต = (Baseโ€˜๐ด)    &   (๐œ‘ โ†’ ๐‘ โˆˆ Fin)    &   (๐œ‘ โ†’ ๐‘… โˆˆ ๐‘‰)    &   ((๐œ‘ โˆง ๐‘ฅ โˆˆ ๐‘ โˆง ๐‘ฆ โˆˆ ๐‘) โ†’ ๐ถ โˆˆ ๐พ)    โ‡’   (๐œ‘ โ†’ (๐‘ฅ โˆˆ ๐‘, ๐‘ฆ โˆˆ ๐‘ โ†ฆ ๐ถ) โˆˆ ๐ต)
 
Theoremeqmat 22147* Two square matrices of the same dimension are equal if they have the same entries. (Contributed by AV, 25-Sep-2019.)
๐ด = (๐‘ Mat ๐‘…)    &   ๐ต = (Baseโ€˜๐ด)    โ‡’   ((๐‘‹ โˆˆ ๐ต โˆง ๐‘Œ โˆˆ ๐ต) โ†’ (๐‘‹ = ๐‘Œ โ†” โˆ€๐‘– โˆˆ ๐‘ โˆ€๐‘— โˆˆ ๐‘ (๐‘–๐‘‹๐‘—) = (๐‘–๐‘Œ๐‘—)))
 
Theoremmatecl 22148 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.)
๐ด = (๐‘ Mat ๐‘…)    &   ๐พ = (Baseโ€˜๐‘…)    โ‡’   ((๐ผ โˆˆ ๐‘ โˆง ๐ฝ โˆˆ ๐‘ โˆง ๐‘€ โˆˆ (Baseโ€˜๐ด)) โ†’ (๐ผ๐‘€๐ฝ) โˆˆ ๐พ)
 
Theoremmatecld 22149 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.)
๐ด = (๐‘ Mat ๐‘…)    &   ๐พ = (Baseโ€˜๐‘…)    &   ๐ต = (Baseโ€˜๐ด)    &   (๐œ‘ โ†’ ๐ผ โˆˆ ๐‘)    &   (๐œ‘ โ†’ ๐ฝ โˆˆ ๐‘)    &   (๐œ‘ โ†’ ๐‘€ โˆˆ ๐ต)    โ‡’   (๐œ‘ โ†’ (๐ผ๐‘€๐ฝ) โˆˆ ๐พ)
 
Theoremmatplusg2 22150 Addition in the matrix ring is cell-wise. (Contributed by Stefan O'Rear, 5-Sep-2015.)
๐ด = (๐‘ Mat ๐‘…)    &   ๐ต = (Baseโ€˜๐ด)    &    โœš = (+gโ€˜๐ด)    &    + = (+gโ€˜๐‘…)    โ‡’   ((๐‘‹ โˆˆ ๐ต โˆง ๐‘Œ โˆˆ ๐ต) โ†’ (๐‘‹ โœš ๐‘Œ) = (๐‘‹ โˆ˜f + ๐‘Œ))
 
Theoremmatvsca2 22151 Scalar multiplication in the matrix ring is cell-wise. (Contributed by Stefan O'Rear, 5-Sep-2015.)
๐ด = (๐‘ Mat ๐‘…)    &   ๐ต = (Baseโ€˜๐ด)    &   ๐พ = (Baseโ€˜๐‘…)    &    ยท = ( ยท๐‘  โ€˜๐ด)    &    ร— = (.rโ€˜๐‘…)    &   ๐ถ = (๐‘ ร— ๐‘)    โ‡’   ((๐‘‹ โˆˆ ๐พ โˆง ๐‘Œ โˆˆ ๐ต) โ†’ (๐‘‹ ยท ๐‘Œ) = ((๐ถ ร— {๐‘‹}) โˆ˜f ร— ๐‘Œ))
 
Theoremmatlmod 22152 The matrix ring is a linear structure. (Contributed by Stefan O'Rear, 4-Sep-2015.)
๐ด = (๐‘ Mat ๐‘…)    โ‡’   ((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ Ring) โ†’ ๐ด โˆˆ LMod)
 
Theoremmatgrp 22153 The matrix ring is a group. (Contributed by AV, 21-Dec-2019.)
๐ด = (๐‘ Mat ๐‘…)    โ‡’   ((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ Ring) โ†’ ๐ด โˆˆ Grp)
 
Theoremmatvscl 22154 Closure of the scalar multiplication in the matrix ring. (lmodvscl 20633 analog.) (Contributed by AV, 27-Nov-2019.)
๐พ = (Baseโ€˜๐‘…)    &   ๐ด = (๐‘ Mat ๐‘…)    &   ๐ต = (Baseโ€˜๐ด)    &    ยท = ( ยท๐‘  โ€˜๐ด)    โ‡’   (((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ Ring) โˆง (๐ถ โˆˆ ๐พ โˆง ๐‘‹ โˆˆ ๐ต)) โ†’ (๐ถ ยท ๐‘‹) โˆˆ ๐ต)
 
Theoremmatsubg 22155 The matrix ring has the same addition as its underlying group. (Contributed by AV, 2-Aug-2019.)
๐ด = (๐‘ Mat ๐‘…)    &   ๐บ = (๐‘… freeLMod (๐‘ ร— ๐‘))    โ‡’   ((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ ๐‘‰) โ†’ (-gโ€˜๐บ) = (-gโ€˜๐ด))
 
Theoremmatplusgcell 22156 Addition in the matrix ring is cell-wise. (Contributed by AV, 2-Aug-2019.)
๐ด = (๐‘ Mat ๐‘…)    &   ๐ต = (Baseโ€˜๐ด)    &    โœš = (+gโ€˜๐ด)    &    + = (+gโ€˜๐‘…)    โ‡’   (((๐‘‹ โˆˆ ๐ต โˆง ๐‘Œ โˆˆ ๐ต) โˆง (๐ผ โˆˆ ๐‘ โˆง ๐ฝ โˆˆ ๐‘)) โ†’ (๐ผ(๐‘‹ โœš ๐‘Œ)๐ฝ) = ((๐ผ๐‘‹๐ฝ) + (๐ผ๐‘Œ๐ฝ)))
 
Theoremmatsubgcell 22157 Subtraction in the matrix ring is cell-wise. (Contributed by AV, 2-Aug-2019.)
๐ด = (๐‘ Mat ๐‘…)    &   ๐ต = (Baseโ€˜๐ด)    &   ๐‘† = (-gโ€˜๐ด)    &    โˆ’ = (-gโ€˜๐‘…)    โ‡’   ((๐‘… โˆˆ Ring โˆง (๐‘‹ โˆˆ ๐ต โˆง ๐‘Œ โˆˆ ๐ต) โˆง (๐ผ โˆˆ ๐‘ โˆง ๐ฝ โˆˆ ๐‘)) โ†’ (๐ผ(๐‘‹๐‘†๐‘Œ)๐ฝ) = ((๐ผ๐‘‹๐ฝ) โˆ’ (๐ผ๐‘Œ๐ฝ)))
 
Theoremmatinvgcell 22158 Additive inversion in the matrix ring is cell-wise. (Contributed by AV, 17-Nov-2019.)
๐ด = (๐‘ Mat ๐‘…)    &   ๐ต = (Baseโ€˜๐ด)    &   ๐‘‰ = (invgโ€˜๐‘…)    &   ๐‘Š = (invgโ€˜๐ด)    โ‡’   ((๐‘… โˆˆ Ring โˆง ๐‘‹ โˆˆ ๐ต โˆง (๐ผ โˆˆ ๐‘ โˆง ๐ฝ โˆˆ ๐‘)) โ†’ (๐ผ(๐‘Šโ€˜๐‘‹)๐ฝ) = (๐‘‰โ€˜(๐ผ๐‘‹๐ฝ)))
 
Theoremmatvscacell 22159 Scalar multiplication in the matrix ring is cell-wise. (Contributed by AV, 7-Aug-2019.)
๐ด = (๐‘ Mat ๐‘…)    &   ๐ต = (Baseโ€˜๐ด)    &   ๐พ = (Baseโ€˜๐‘…)    &    ยท = ( ยท๐‘  โ€˜๐ด)    &    ร— = (.rโ€˜๐‘…)    โ‡’   ((๐‘… โˆˆ Ring โˆง (๐‘‹ โˆˆ ๐พ โˆง ๐‘Œ โˆˆ ๐ต) โˆง (๐ผ โˆˆ ๐‘ โˆง ๐ฝ โˆˆ ๐‘)) โ†’ (๐ผ(๐‘‹ ยท ๐‘Œ)๐ฝ) = (๐‘‹ ร— (๐ผ๐‘Œ๐ฝ)))
 
Theoremmatgsum 22160* Finite commutative sums in a matrix algebra are taken componentwise. (Contributed by AV, 26-Sep-2019.)
๐ด = (๐‘ Mat ๐‘…)    &   ๐ต = (Baseโ€˜๐ด)    &    0 = (0gโ€˜๐ด)    &   (๐œ‘ โ†’ ๐‘ โˆˆ Fin)    &   (๐œ‘ โ†’ ๐ฝ โˆˆ ๐‘Š)    &   (๐œ‘ โ†’ ๐‘… โˆˆ Ring)    &   ((๐œ‘ โˆง ๐‘ฆ โˆˆ ๐ฝ) โ†’ (๐‘– โˆˆ ๐‘, ๐‘— โˆˆ ๐‘ โ†ฆ ๐‘ˆ) โˆˆ ๐ต)    &   (๐œ‘ โ†’ (๐‘ฆ โˆˆ ๐ฝ โ†ฆ (๐‘– โˆˆ ๐‘, ๐‘— โˆˆ ๐‘ โ†ฆ ๐‘ˆ)) finSupp 0 )    โ‡’   (๐œ‘ โ†’ (๐ด ฮฃg (๐‘ฆ โˆˆ ๐ฝ โ†ฆ (๐‘– โˆˆ ๐‘, ๐‘— โˆˆ ๐‘ โ†ฆ ๐‘ˆ))) = (๐‘– โˆˆ ๐‘, ๐‘— โˆˆ ๐‘ โ†ฆ (๐‘… ฮฃg (๐‘ฆ โˆˆ ๐ฝ โ†ฆ ๐‘ˆ))))
 
11.4.3  The matrix algebra

The main result of this subsection are the theorems showing that (๐‘ Mat ๐‘…) is a ring (see matring 22166) and an associative algebra (see matassa 22167). Additionally, theorems for the identity matrix and transposed matrices are provided.

 
Theoremmatmulr 22161 Multiplication in the matrix algebra. (Contributed by Stefan O'Rear, 4-Sep-2015.)
๐ด = (๐‘ Mat ๐‘…)    &    ยท = (๐‘… maMul โŸจ๐‘, ๐‘, ๐‘โŸฉ)    โ‡’   ((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ ๐‘‰) โ†’ ยท = (.rโ€˜๐ด))
 
Theoremmamumat1cl 22162* The identity matrix (as operation in maps-to notation) is a matrix. (Contributed by Stefan O'Rear, 2-Sep-2015.)
๐ต = (Baseโ€˜๐‘…)    &   (๐œ‘ โ†’ ๐‘… โˆˆ Ring)    &    1 = (1rโ€˜๐‘…)    &    0 = (0gโ€˜๐‘…)    &   ๐ผ = (๐‘– โˆˆ ๐‘€, ๐‘— โˆˆ ๐‘€ โ†ฆ if(๐‘– = ๐‘—, 1 , 0 ))    &   (๐œ‘ โ†’ ๐‘€ โˆˆ Fin)    โ‡’   (๐œ‘ โ†’ ๐ผ โˆˆ (๐ต โ†‘m (๐‘€ ร— ๐‘€)))
 
Theoremmat1comp 22163* The components of the identity matrix (as operation in maps-to notation). (Contributed by AV, 22-Jul-2019.)
๐ต = (Baseโ€˜๐‘…)    &   (๐œ‘ โ†’ ๐‘… โˆˆ Ring)    &    1 = (1rโ€˜๐‘…)    &    0 = (0gโ€˜๐‘…)    &   ๐ผ = (๐‘– โˆˆ ๐‘€, ๐‘— โˆˆ ๐‘€ โ†ฆ if(๐‘– = ๐‘—, 1 , 0 ))    &   (๐œ‘ โ†’ ๐‘€ โˆˆ Fin)    โ‡’   ((๐ด โˆˆ ๐‘€ โˆง ๐ฝ โˆˆ ๐‘€) โ†’ (๐ด๐ผ๐ฝ) = if(๐ด = ๐ฝ, 1 , 0 ))
 
Theoremmamulid 22164* 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.)
๐ต = (Baseโ€˜๐‘…)    &   (๐œ‘ โ†’ ๐‘… โˆˆ Ring)    &    1 = (1rโ€˜๐‘…)    &    0 = (0gโ€˜๐‘…)    &   ๐ผ = (๐‘– โˆˆ ๐‘€, ๐‘— โˆˆ ๐‘€ โ†ฆ if(๐‘– = ๐‘—, 1 , 0 ))    &   (๐œ‘ โ†’ ๐‘€ โˆˆ Fin)    &   (๐œ‘ โ†’ ๐‘ โˆˆ Fin)    &   ๐น = (๐‘… maMul โŸจ๐‘€, ๐‘€, ๐‘โŸฉ)    &   (๐œ‘ โ†’ ๐‘‹ โˆˆ (๐ต โ†‘m (๐‘€ ร— ๐‘)))    โ‡’   (๐œ‘ โ†’ (๐ผ๐น๐‘‹) = ๐‘‹)
 
Theoremmamurid 22165* 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.)
๐ต = (Baseโ€˜๐‘…)    &   (๐œ‘ โ†’ ๐‘… โˆˆ Ring)    &    1 = (1rโ€˜๐‘…)    &    0 = (0gโ€˜๐‘…)    &   ๐ผ = (๐‘– โˆˆ ๐‘€, ๐‘— โˆˆ ๐‘€ โ†ฆ if(๐‘– = ๐‘—, 1 , 0 ))    &   (๐œ‘ โ†’ ๐‘€ โˆˆ Fin)    &   (๐œ‘ โ†’ ๐‘ โˆˆ Fin)    &   ๐น = (๐‘… maMul โŸจ๐‘, ๐‘€, ๐‘€โŸฉ)    &   (๐œ‘ โ†’ ๐‘‹ โˆˆ (๐ต โ†‘m (๐‘ ร— ๐‘€)))    โ‡’   (๐œ‘ โ†’ (๐‘‹๐น๐ผ) = ๐‘‹)
 
Theoremmatring 22166 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.)
๐ด = (๐‘ Mat ๐‘…)    โ‡’   ((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ Ring) โ†’ ๐ด โˆˆ Ring)
 
Theoremmatassa 22167 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.)
๐ด = (๐‘ Mat ๐‘…)    โ‡’   ((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ CRing) โ†’ ๐ด โˆˆ AssAlg)
 
Theoremmatmulcell 22168* Multiplication in the matrix ring for a single cell of a matrix. (Contributed by AV, 17-Nov-2019.) (Revised by AV, 3-Jul-2022.)
๐ด = (๐‘ Mat ๐‘…)    &   ๐ต = (Baseโ€˜๐ด)    &    ร— = (.rโ€˜๐ด)    โ‡’   ((๐‘… โˆˆ Ring โˆง (๐‘‹ โˆˆ ๐ต โˆง ๐‘Œ โˆˆ ๐ต) โˆง (๐ผ โˆˆ ๐‘ โˆง ๐ฝ โˆˆ ๐‘)) โ†’ (๐ผ(๐‘‹ ร— ๐‘Œ)๐ฝ) = (๐‘… ฮฃg (๐‘— โˆˆ ๐‘ โ†ฆ ((๐ผ๐‘‹๐‘—)(.rโ€˜๐‘…)(๐‘—๐‘Œ๐ฝ)))))
 
Theoremmpomatmul 22169* Multiplication of two N x N matrices given in maps-to notation. (Contributed by AV, 29-Oct-2019.)
๐ด = (๐‘ Mat ๐‘…)    &   ๐ต = (Baseโ€˜๐‘…)    &    ร— = (.rโ€˜๐ด)    &    ยท = (.rโ€˜๐‘…)    &   (๐œ‘ โ†’ ๐‘… โˆˆ ๐‘‰)    &   (๐œ‘ โ†’ ๐‘ โˆˆ Fin)    &   ๐‘‹ = (๐‘– โˆˆ ๐‘, ๐‘— โˆˆ ๐‘ โ†ฆ ๐ถ)    &   ๐‘Œ = (๐‘– โˆˆ ๐‘, ๐‘— โˆˆ ๐‘ โ†ฆ ๐ธ)    &   ((๐œ‘ โˆง ๐‘– โˆˆ ๐‘ โˆง ๐‘— โˆˆ ๐‘) โ†’ ๐ถ โˆˆ ๐ต)    &   ((๐œ‘ โˆง ๐‘– โˆˆ ๐‘ โˆง ๐‘— โˆˆ ๐‘) โ†’ ๐ธ โˆˆ ๐ต)    &   ((๐œ‘ โˆง (๐‘˜ = ๐‘– โˆง ๐‘š = ๐‘—)) โ†’ ๐ท = ๐ถ)    &   ((๐œ‘ โˆง (๐‘š = ๐‘– โˆง ๐‘™ = ๐‘—)) โ†’ ๐น = ๐ธ)    &   ((๐œ‘ โˆง ๐‘˜ โˆˆ ๐‘ โˆง ๐‘š โˆˆ ๐‘) โ†’ ๐ท โˆˆ ๐‘ˆ)    &   ((๐œ‘ โˆง ๐‘š โˆˆ ๐‘ โˆง ๐‘™ โˆˆ ๐‘) โ†’ ๐น โˆˆ ๐‘Š)    โ‡’   (๐œ‘ โ†’ (๐‘‹ ร— ๐‘Œ) = (๐‘˜ โˆˆ ๐‘, ๐‘™ โˆˆ ๐‘ โ†ฆ (๐‘… ฮฃg (๐‘š โˆˆ ๐‘ โ†ฆ (๐ท ยท ๐น)))))
 
Theoremmat1 22170* 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.)
๐ด = (๐‘ Mat ๐‘…)    &    1 = (1rโ€˜๐‘…)    &    0 = (0gโ€˜๐‘…)    โ‡’   ((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ Ring) โ†’ (1rโ€˜๐ด) = (๐‘– โˆˆ ๐‘, ๐‘— โˆˆ ๐‘ โ†ฆ if(๐‘– = ๐‘—, 1 , 0 )))
 
Theoremmat1ov 22171 Entries of an identity matrix, deduction form. (Contributed by Stefan O'Rear, 10-Jul-2018.)
๐ด = (๐‘ Mat ๐‘…)    &    1 = (1rโ€˜๐‘…)    &    0 = (0gโ€˜๐‘…)    &   (๐œ‘ โ†’ ๐‘ โˆˆ Fin)    &   (๐œ‘ โ†’ ๐‘… โˆˆ Ring)    &   (๐œ‘ โ†’ ๐ผ โˆˆ ๐‘)    &   (๐œ‘ โ†’ ๐ฝ โˆˆ ๐‘)    &   ๐‘ˆ = (1rโ€˜๐ด)    โ‡’   (๐œ‘ โ†’ (๐ผ๐‘ˆ๐ฝ) = if(๐ผ = ๐ฝ, 1 , 0 ))
 
Theoremmat1bas 22172 The identity matrix is a matrix. (Contributed by AV, 15-Feb-2019.)
๐ด = (๐‘ Mat ๐‘…)    &   ๐ต = (Baseโ€˜๐ด)    &    1 = (1rโ€˜(๐‘ Mat ๐‘…))    โ‡’   ((๐‘… โˆˆ Ring โˆง ๐‘ โˆˆ Fin) โ†’ 1 โˆˆ ๐ต)
 
Theoremmatsc 22173* The identity matrix multiplied with a scalar. (Contributed by Stefan O'Rear, 16-Jul-2018.)
๐ด = (๐‘ Mat ๐‘…)    &   ๐พ = (Baseโ€˜๐‘…)    &    ยท = ( ยท๐‘  โ€˜๐ด)    &    0 = (0gโ€˜๐‘…)    โ‡’   ((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ Ring โˆง ๐ฟ โˆˆ ๐พ) โ†’ (๐ฟ ยท (1rโ€˜๐ด)) = (๐‘– โˆˆ ๐‘, ๐‘— โˆˆ ๐‘ โ†ฆ if(๐‘– = ๐‘—, ๐ฟ, 0 )))
 
Theoremofco2 22174 Distribution law for the function operation and the composition of functions. (Contributed by Stefan O'Rear, 17-Jul-2018.)
(((๐น โˆˆ V โˆง ๐บ โˆˆ V) โˆง (Fun ๐ป โˆง (๐น โˆ˜ ๐ป) โˆˆ V โˆง (๐บ โˆ˜ ๐ป) โˆˆ V)) โ†’ ((๐น โˆ˜f ๐‘…๐บ) โˆ˜ ๐ป) = ((๐น โˆ˜ ๐ป) โˆ˜f ๐‘…(๐บ โˆ˜ ๐ป)))
 
Theoremoftpos 22175 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.)
((๐น โˆˆ ๐‘‰ โˆง ๐บ โˆˆ ๐‘Š) โ†’ tpos (๐น โˆ˜f ๐‘…๐บ) = (tpos ๐น โˆ˜f ๐‘…tpos ๐บ))
 
Theoremmattposcl 22176 The transpose of a square matrix is a square matrix of the same size. (Contributed by SO, 9-Jul-2018.)
๐ด = (๐‘ Mat ๐‘…)    &   ๐ต = (Baseโ€˜๐ด)    โ‡’   (๐‘€ โˆˆ ๐ต โ†’ tpos ๐‘€ โˆˆ ๐ต)
 
Theoremmattpostpos 22177 The transpose of the transpose of a square matrix is the square matrix itself. (Contributed by SO, 17-Jul-2018.)
๐ด = (๐‘ Mat ๐‘…)    &   ๐ต = (Baseโ€˜๐ด)    โ‡’   (๐‘€ โˆˆ ๐ต โ†’ tpos tpos ๐‘€ = ๐‘€)
 
Theoremmattposvs 22178 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.)
๐ด = (๐‘ Mat ๐‘…)    &   ๐ต = (Baseโ€˜๐ด)    &   ๐พ = (Baseโ€˜๐‘…)    &    ยท = ( ยท๐‘  โ€˜๐ด)    โ‡’   ((๐‘‹ โˆˆ ๐พ โˆง ๐‘Œ โˆˆ ๐ต) โ†’ tpos (๐‘‹ ยท ๐‘Œ) = (๐‘‹ ยท tpos ๐‘Œ))
 
Theoremmattpos1 22179 The transposition of the identity matrix is the identity matrix. (Contributed by Stefan O'Rear, 17-Jul-2018.)
๐ด = (๐‘ Mat ๐‘…)    &    1 = (1rโ€˜๐ด)    โ‡’   ((๐‘ โˆˆ Fin โˆง ๐‘… โˆˆ Ring) โ†’ tpos 1 = 1 )
 
Theoremtposmap 22180 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.)
(๐ด โˆˆ (๐ต โ†‘m (๐ผ ร— ๐ฝ)) โ†’ tpos ๐ด โˆˆ (๐ต โ†‘m (๐ฝ ร— ๐ผ)))
 
Theoremmamutpos 22181 Behavior of transposes in matrix products, see also the statement in [Lang] p. 505. (Contributed by Stefan O'Rear, 9-Jul-2018.)
๐น = (๐‘… maMul โŸจ๐‘€, ๐‘, ๐‘ƒโŸฉ)    &   ๐บ = (๐‘… maMul โŸจ๐‘ƒ, ๐‘, ๐‘€โŸฉ)    &   ๐ต = (Baseโ€˜๐‘…)    &   (๐œ‘ โ†’ ๐‘… โˆˆ CRing)    &   (๐œ‘ โ†’ ๐‘€ โˆˆ Fin)    &   (๐œ‘ โ†’ ๐‘ โˆˆ Fin)    &   (๐œ‘ โ†’ ๐‘ƒ โˆˆ Fin)    &   (๐œ‘ โ†’ ๐‘‹ โˆˆ (๐ต โ†‘m (๐‘€ ร— ๐‘)))    &   (๐œ‘ โ†’ ๐‘Œ โˆˆ (๐ต โ†‘m (๐‘ ร— ๐‘ƒ)))    โ‡’   (๐œ‘ โ†’ tpos (๐‘‹๐น๐‘Œ) = (tpos ๐‘Œ๐บtpos ๐‘‹))
 
Theoremmattposm 22182 Multiplying two transposed matrices results in the transposition of the product of the two matrices. (Contributed by Stefan O'Rear, 17-Jul-2018.)
๐ด = (๐‘ Mat ๐‘…)    &   ๐ต = (Baseโ€˜๐ด)    &    ยท = (.rโ€˜๐ด)    โ‡’   ((๐‘… โˆˆ CRing โˆง ๐‘‹ โˆˆ ๐ต โˆง ๐‘Œ โˆˆ ๐ต) โ†’ tpos (๐‘‹ ยท ๐‘Œ) = (tpos ๐‘Œ ยท tpos ๐‘‹))
 
Theoremmatgsumcl 22183* 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.)
๐ด = (๐‘ Mat ๐‘…)    &   ๐ต = (Baseโ€˜๐ด)    &   ๐‘ˆ = (mulGrpโ€˜๐‘…)    โ‡’   ((๐‘… โˆˆ CRing โˆง ๐‘€ โˆˆ ๐ต) โ†’ (๐‘ˆ ฮฃg (๐‘Ÿ โˆˆ ๐‘ โ†ฆ (๐‘Ÿ๐‘€๐‘Ÿ))) โˆˆ (Baseโ€˜๐‘…))
 
Theoremmadetsumid 22184* The identity summand in the Leibniz' formula of a determinant for a square matrix over a commutative ring. (Contributed by AV, 29-Dec-2018.)
๐ด = (๐‘ Mat ๐‘…)    &   ๐ต = (Baseโ€˜๐ด)    &   ๐‘ˆ = (mulGrpโ€˜๐‘…)    &   ๐‘Œ = (โ„คRHomโ€˜๐‘…)    &   ๐‘† = (pmSgnโ€˜๐‘)    &    ยท = (.rโ€˜๐‘…)    โ‡’   ((๐‘… โˆˆ CRing โˆง ๐‘€ โˆˆ ๐ต โˆง ๐‘ƒ = ( I โ†พ ๐‘)) โ†’ (((๐‘Œ โˆ˜ ๐‘†)โ€˜๐‘ƒ) ยท (๐‘ˆ ฮฃg (๐‘Ÿ โˆˆ ๐‘ โ†ฆ ((๐‘ƒโ€˜๐‘Ÿ)๐‘€๐‘Ÿ)))) = (๐‘ˆ ฮฃg (๐‘Ÿ โˆˆ ๐‘ โ†ฆ (๐‘Ÿ๐‘€๐‘Ÿ))))
 
Theoremmatepmcl 22185* 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.)
๐ด = (๐‘ Mat ๐‘…)    &   ๐ต = (Baseโ€˜๐ด)    &   ๐‘ƒ = (Baseโ€˜(SymGrpโ€˜๐‘))    โ‡’   ((๐‘… โˆˆ Ring โˆง ๐‘„ โˆˆ ๐‘ƒ โˆง ๐‘€ โˆˆ ๐ต) โ†’ โˆ€๐‘› โˆˆ ๐‘ ((๐‘„โ€˜๐‘›)๐‘€๐‘›) โˆˆ (Baseโ€˜๐‘…))
 
Theoremmatepm2cl 22186* 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.)
๐ด = (๐‘ Mat ๐‘…)    &   ๐ต = (Baseโ€˜๐ด)    &   ๐‘ƒ = (Baseโ€˜(SymGrpโ€˜๐‘))    โ‡’   ((๐‘… โˆˆ Ring โˆง ๐‘„ โˆˆ ๐‘ƒ โˆง ๐‘€ โˆˆ ๐ต) โ†’ โˆ€๐‘› โˆˆ ๐‘ (๐‘›๐‘€(๐‘„โ€˜๐‘›)) โˆˆ (Baseโ€˜๐‘…))
 
Theoremmadetsmelbas 22187* A summand of the determinant of a matrix belongs to the underlying ring. (Contributed by AV, 1-Jan-2019.)
๐‘ƒ = (Baseโ€˜(SymGrpโ€˜๐‘))    &   ๐‘† = (pmSgnโ€˜๐‘)    &   ๐‘Œ = (โ„คRHomโ€˜๐‘…)    &   ๐ด = (๐‘ Mat ๐‘…)    &   ๐ต = (Baseโ€˜๐ด)    &   ๐บ = (mulGrpโ€˜๐‘…)    โ‡’   ((๐‘… โˆˆ CRing โˆง ๐‘€ โˆˆ ๐ต โˆง ๐‘„ โˆˆ ๐‘ƒ) โ†’ (((๐‘Œ โˆ˜ ๐‘†)โ€˜๐‘„)(.rโ€˜๐‘…)(๐บ ฮฃg (๐‘› โˆˆ ๐‘ โ†ฆ ((๐‘„โ€˜๐‘›)๐‘€๐‘›)))) โˆˆ (Baseโ€˜๐‘…))
 
Theoremmadetsmelbas2 22188* A summand of the determinant of a matrix belongs to the underlying ring. (Contributed by AV, 1-Jan-2019.)
๐‘ƒ = (Baseโ€˜(SymGrpโ€˜๐‘))    &   ๐‘† = (pmSgnโ€˜๐‘)    &   ๐‘Œ = (โ„คRHomโ€˜๐‘…)    &   ๐ด = (๐‘ Mat ๐‘…)    &   ๐ต = (Baseโ€˜๐ด)    &   ๐บ = (mulGrpโ€˜๐‘…)    โ‡’   ((๐‘… โˆˆ CRing โˆง ๐‘€ โˆˆ ๐ต โˆง ๐‘„ โˆˆ ๐‘ƒ) โ†’ (((๐‘Œ โˆ˜ ๐‘†)โ€˜๐‘„)(.rโ€˜๐‘…)(๐บ ฮฃg (๐‘› โˆˆ ๐‘ โ†ฆ (๐‘›๐‘€(๐‘„โ€˜๐‘›))))) โˆˆ (Baseโ€˜๐‘…))
 
11.4.4  Matrices of dimension 0 and 1

As already mentioned before, and shown in mat0dimbas0 22189, the empty set is the sole zero-dimensional matrix (also called "empty matrix", see Wikipedia https://en.wikipedia.org/wiki/Matrix_(mathematics)#Empty_matrices). 22189 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 22193.

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 22210.

 
Theoremmat0dimbas0 22189 The empty set is the one and only matrix of dimension 0, called "the empty matrix". (Contributed by AV, 27-Feb-2019.)
(๐‘… โˆˆ ๐‘‰ โ†’ (Baseโ€˜(โˆ… Mat ๐‘…)) = {โˆ…})
 
Theoremmat0dim0 22190 The zero of the algebra of matrices with dimension 0. (Contributed by AV, 6-Aug-2019.)
๐ด = (โˆ… Mat ๐‘…)    โ‡’   (๐‘… โˆˆ Ring โ†’ (0gโ€˜๐ด) = โˆ…)
 
Theoremmat0dimid 22191 The identity of the algebra of matrices with dimension 0. (Contributed by AV, 6-Aug-2019.)
๐ด = (โˆ… Mat ๐‘…)    โ‡’   (๐‘… โˆˆ Ring โ†’ (1rโ€˜๐ด) = โˆ…)
 
Theoremmat0dimscm 22192 The scalar multiplication in the algebra of matrices with dimension 0. (Contributed by AV, 6-Aug-2019.)
๐ด = (โˆ… Mat ๐‘…)    โ‡’   ((๐‘… โˆˆ Ring โˆง ๐‘‹ โˆˆ (Baseโ€˜๐‘…)) โ†’ (๐‘‹( ยท๐‘  โ€˜๐ด)โˆ…) = โˆ…)
 
Theoremmat0dimcrng 22193 The algebra of matrices with dimension 0 (over an arbitrary ring!) is a commutative ring. (Contributed by AV, 10-Aug-2019.)
๐ด = (โˆ… Mat ๐‘…)    โ‡’   (๐‘… โˆˆ Ring โ†’ ๐ด โˆˆ CRing)
 
Theoremmat1dimelbas 22194* 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.)
๐ด = ({๐ธ} Mat ๐‘…)    &   ๐ต = (Baseโ€˜๐‘…)    &   ๐‘‚ = โŸจ๐ธ, ๐ธโŸฉ    โ‡’   ((๐‘… โˆˆ Ring โˆง ๐ธ โˆˆ ๐‘‰) โ†’ (๐‘€ โˆˆ (Baseโ€˜๐ด) โ†” โˆƒ๐‘Ÿ โˆˆ ๐ต ๐‘€ = {โŸจ๐‘‚, ๐‘ŸโŸฉ}))
 
Theoremmat1dimbas 22195 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.)
๐ด = ({๐ธ} Mat ๐‘…)    &   ๐ต = (Baseโ€˜๐‘…)    &   ๐‘‚ = โŸจ๐ธ, ๐ธโŸฉ    โ‡’   ((๐‘… โˆˆ Ring โˆง ๐ธ โˆˆ ๐‘‰ โˆง ๐‘‹ โˆˆ ๐ต) โ†’ {โŸจ๐‘‚, ๐‘‹โŸฉ} โˆˆ (Baseโ€˜๐ด))
 
Theoremmat1dim0 22196 The zero of the algebra of matrices with dimension 1. (Contributed by AV, 15-Aug-2019.)
๐ด = ({๐ธ} Mat ๐‘…)    &   ๐ต = (Baseโ€˜๐‘…)    &   ๐‘‚ = โŸจ๐ธ, ๐ธโŸฉ    โ‡’   ((๐‘… โˆˆ Ring โˆง ๐ธ โˆˆ ๐‘‰) โ†’ (0gโ€˜๐ด) = {โŸจ๐‘‚, (0gโ€˜๐‘…)โŸฉ})
 
Theoremmat1dimid 22197 The identity of the algebra of matrices with dimension 1. (Contributed by AV, 15-Aug-2019.)
๐ด = ({๐ธ} Mat ๐‘…)    &   ๐ต = (Baseโ€˜๐‘…)    &   ๐‘‚ = โŸจ๐ธ, ๐ธโŸฉ    โ‡’   ((๐‘… โˆˆ Ring โˆง ๐ธ โˆˆ ๐‘‰) โ†’ (1rโ€˜๐ด) = {โŸจ๐‘‚, (1rโ€˜๐‘…)โŸฉ})
 
Theoremmat1dimscm 22198 The scalar multiplication in the algebra of matrices with dimension 1. (Contributed by AV, 16-Aug-2019.)
๐ด = ({๐ธ} Mat ๐‘…)    &   ๐ต = (Baseโ€˜๐‘…)    &   ๐‘‚ = โŸจ๐ธ, ๐ธโŸฉ    โ‡’   (((๐‘… โˆˆ Ring โˆง ๐ธ โˆˆ ๐‘‰) โˆง (๐‘‹ โˆˆ ๐ต โˆง ๐‘Œ โˆˆ ๐ต)) โ†’ (๐‘‹( ยท๐‘  โ€˜๐ด){โŸจ๐‘‚, ๐‘ŒโŸฉ}) = {โŸจ๐‘‚, (๐‘‹(.rโ€˜๐‘…)๐‘Œ)โŸฉ})
 
Theoremmat1dimmul 22199 The ring multiplication in the algebra of matrices with dimension 1. (Contributed by AV, 16-Aug-2019.) (Proof shortened by AV, 18-Apr-2021.)
๐ด = ({๐ธ} Mat ๐‘…)    &   ๐ต = (Baseโ€˜๐‘…)    &   ๐‘‚ = โŸจ๐ธ, ๐ธโŸฉ    โ‡’   (((๐‘… โˆˆ Ring โˆง ๐ธ โˆˆ ๐‘‰) โˆง (๐‘‹ โˆˆ ๐ต โˆง ๐‘Œ โˆˆ ๐ต)) โ†’ ({โŸจ๐‘‚, ๐‘‹โŸฉ} (.rโ€˜๐ด){โŸจ๐‘‚, ๐‘ŒโŸฉ}) = {โŸจ๐‘‚, (๐‘‹(.rโ€˜๐‘…)๐‘Œ)โŸฉ})
 
Theoremmat1dimcrng 22200 The algebra of matrices with dimension 1 over a commutative ring is a commutative ring. (Contributed by AV, 16-Aug-2019.)
๐ด = ({๐ธ} Mat ๐‘…)    &   ๐ต = (Baseโ€˜๐‘…)    &   ๐‘‚ = โŸจ๐ธ, ๐ธโŸฉ    โ‡’   ((๐‘… โˆˆ CRing โˆง ๐ธ โˆˆ ๐‘‰) โ†’ ๐ด โˆˆ CRing)
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78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-47000 471 47001-47100 472 47101-47200 473 47201-47300 474 47301-47400 475 47401-47500 476 47501-47600 477 47601-47700 478 47701-47800 479 47801-47900 480 47901-47940
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