Remark

It should be noted that a matrix for which the indexing sets of the rows and columns have the same cardinal but are not identical, must not be considered as a square matrix; in particular, the product of two such matrices over a ring is not defined.

Clearly addition and multiplication of square matrices over A with a finite set as indexing set of the rows and columns, define on the set of these matrices a ring structure because of formulae (7), (8) and (9) (no. 3); the matrix $(\delta_{ij})$, where $\delta_{ij}$ is the Kronecker index (for $i \in I, j \in I$), is the unit element of this ring and is denoted by $I_n$ or $l_n$ when I has $n$ elements. When $I = \{1, n\}$, the ring of matrices thus defined is denoted simply by $\mathbf{M}_n(A)$; the group of invertible elements of $\mathbf{M}_n(A)$ is denoted by $\mathbf{GL}_n(A)$ or $\mathbf{GL}(n, A)$.

For a square matrix $U = (a_{ij})$ of order $n$ over A to be right (resp. left) invertible, it is necessary and sufficient that, for every system $(b_i)_{1 \leq i \leq n}$ of elements of A, the system of $n$ equations in $n$ unknowns

(resp. $\sum_{j=1}^n x_j a_{ji} = b_i$)

have one solution $(x_i)$ in A.

Let I be a finite indexing set, A a ring and E a right (resp. left) A-module with basis $(e_i)_{i \in I}$. For every endomorphism $u$ of E, the matrix $M(u)$ of $u$ with respect to the two bases identical with $(e_i)$ is a square matrix; more briefly, it is called the matrix of $u$ with respect to the basis $(e_i)$.

Suppose that $I = \{1, n\}$. The mapping $u \mapsto M(u)$ (resp. $u \mapsto {}^t M(u)$) is an isomorphism of the ring $\mathrm{End}_A(E)$ onto $\mathbf{M}_n(A)$ (resp. onto the opposite ring of $\mathbf{M}_n(A)$, as follows from formulae (18 D) (resp. 18 G)) (no. 4). The invertible elements of the ring $\mathbf{M}_n(\mathbf{A})$ called invertible matrices, correspond under the mapping $u \mapsto M(u)$ (resp. $u \mapsto {}^t M(u)$) to the automorphisms of E; the group $\mathbf{GL}(n, \mathbf{A})$ is therefore canonically identified with the group $\mathbf{GL}(\mathbf{A}_d^n)$.

If $u$ is an automorphism of E, its contragredient $\check{u}$ is an automorphism of the left (resp. right) A-module $E^*$, such that $\check{u} = ({}^t u)^{-1} = {}^t (u^{-1})$ (§ 2, no. 5, Definition 6); if $M(\check{u})$ is the matrix of $\check{u}$ with respect to the dual basis $(e_i^*)$, then, by virtue of Proposition 3 (no. 4),

For every invertible matrix $X$, it therefore follows that ${}^t (X^{-1}) = ({}^t X)^{-1}$; this matrix is also denoted by ${}^t X^{-1}$ and called the contragredient of the matrix $X$.

Let $\sigma$ be an automorphism of the ring $\mathbf{A}$; for every semi-linear mapping $u : E \to E$ relative to $\sigma$, the matrix $M(u)$ of this mapping with respect to a basis $(e_i)$ of E is also a square matrix. It follows immediately from (27 D) (no. 6) that, if $u$ is bijective, then

Let E be an A-module which is the direct sum of a finite family $(E_i)_{i \in I}$ of submodules; for every endomorphism $u$ of E, the matrix $M(u) = (u_{ki})$ of $u$ with respect to the two decompositions of E identical with $(E_i)$ (no. 5) is a square matrix of linear mappings. In order that $u(E_i) \subset E_i$ for all $i \in I$, it is necessary and sufficient that $u_{ki} = 0$ for $k \neq i$. When $I = \{1, n\}$, the relations

are equivalent to the relations $u_{ki} = 0$ for $k < i$.

Examples of square matrices. I. Diagonal matrices. In a square matrix

the elements both of whose indices are equal are called diagonal elements and the family $(m_{\iota \iota})_{\iota \in I}$ is called the diagonal of $M$; a square matrix $M = (m_{\iota \kappa})$ over a ring, whose elements other than the diagonal elements are zero, is called a diagonal matrix. For every family $(a_\iota)_{\iota \in I}$ of elements of a ring $\mathbf{A}$, the diagonal matrix $(m_{\iota \kappa})$ such that $m_{\iota \iota} = a_\iota$ for all $\iota \in I$ is denoted by $\operatorname{diag}(a_\iota)_{\iota \in I}$ (or $\operatorname{diag}(a_1, a_2, \ldots, a_n)$ when $I = \{1, n\}$). In the set $\mathbf{M}_n(\mathbf{A})$ of square matrices of order $n$ over $\mathbf{A}$, the unit matrix $I_n$ is a diagonal matrix and also every multiple $a I_n = I_n a$ of this matrix by a scalar $a$ (the diagonal matrix (called scalar) all of whose diagonal elements are equal to $a$).

For every family $(d_i)_{1 \leq i \leq n}$ of elements of $\mathbf{A}$ and every matrix $X = (x_{ij})$ of type $(n, q)$ (resp. $(p, n)$) over $\mathbf{A}$, writing $D = \operatorname{diag}(d_i)$,

In particular, for two diagonal matrices of order $n$,

(31)

The diagonal matrices therefore form a subring of $\mathbf{M}_n(\mathbf{A})$ isomorphic to the product ring $\mathbf{A}^n$; the scalar matrices form a subring isomorphic to $\mathbf{A}$.

II. Permutation matrices; monomial matrices. Let $\pi$ be any permutation of a finite set $\mathbf{I}$ and let $(e_i)_{i \in \mathbf{I}}$ be the canonical basis of the $\mathbf{A}$-module $\mathbf{E} = \mathbf{A}_d^\mathbf{I}$; there exists one and only one endomorphism $u_\pi$ of $\mathbf{E}$ such that, for all $i \in \mathbf{I}$, $u_\pi(e_i) = e_{\pi(i)}$ (§ 1, no. 11, Corollary 3 to Proposition 17). For all $i \in \mathbf{I}$, the column of index $i$ in the matrix $M(u_\pi)$ with respect to the basis $(e_i)$ has all its elements zero except the one in the row of index $\pi(i)$, which is equal to 1. By an abuse of language, $M(u_\pi)$ is called the matrix of the permutation $\pi$. It is immediate that for any two permutations $\sigma, \tau$ of $\mathbf{I}$, $u_{\sigma \tau} = u_\sigma \circ u_\tau$ and that for the identity permutation $\varepsilon$, $u_\varepsilon$ is the identity; the mapping $\pi \mapsto M(u_\pi)$ is therefore an isomorphism of the symmetric group $\mathfrak{S}_\mathbf{I}$ onto the group of permutation matrices.

Each row and each column of a permutation matrix contains only a single element $\neq 0$. A finite square matrix $R$ over a non-zero ring $\mathbf{A}$, with this property, is called a monomial matrix; let $r_i$ be the unique element $\neq 0$ in the column of $R$ of index $i$ and let $\pi(i)$ be the index of the row where this element is; clearly $\pi$ is a permutation of the indexing set $\mathbf{I}$ and $R = M(u_\pi)D$, where $D = \text{diag}(r_i)$.

III. Triangular matrices. In the ring $\mathbf{M}_n(\mathbf{A})$ of square matrices of order $n$ over a ring $\mathbf{A}$, any matrix $(a_{ij})$ such that $a_{ij} = 0$ for $i > j$ (resp. $i < j$) is called an upper (resp. lower) triangular matrix; it is also said that such a matrix has only zeros below (resp. above) its diagonal. It is immediately established that the upper (resp. lower) triangular matrices form a subring $S$ (resp. $T$) of $\mathbf{M}_n(\mathbf{A})$, $S \cap T$ being obviously the ring of diagonal matrices.

The set $S'$ (resp. $T'$) of matrices in $S$ (resp. $T$) whose diagonal elements are invertible is a multiplicative group of matrices called the upper (resp. lower) total triangular group, this follows immediately from § 1, no. 11, Remark 5. The set $S_1$ (resp. $T_1$) of matrices in $S$ (resp. $T$) whose diagonal elements are all equal to 1 is a subgroup of the above group, called the upper (resp. lower) strict triangular group, and every matrix $M \in S'$ (resp. $M \in T'$) whose diagonal is $(d_i)$, may be written as $M = DM_1 = M'_1D$, where $D = \text{diag}(d_i)$ and $M_1$ and $M'_1$ matrices belonging to $S_1$ (resp. $T_1$).

IV. Diagonal and triangular matrices of matrices. Let $(I_k)_{1 \leq k \leq p}$ be a partition of the finite set $\mathbf{I}$; every square matrix over a ring $\mathbf{A}$ with indexing set $\mathbf{I}$ can be written in the form of a square matrix of matrices corresponding to the same partition $(\mathbf{I}_k)$ of the indexing set of the rows and the indexing set of the columns (no. 5)

where each $X_{kk}$ is a square matrix with $\mathbf{I}_k$ as indexing set of the rows and columns.

With this notation, (32) will be called a diagonal (resp. upper triangular, resp. lower triangular) matrix of matrices if all the matrices $X_{ij}$ such that $i \neq j$ (resp. $i > j$, resp. $i < j$) are zero. The interpretation of endomorphisms $u$ whose matrix is a diagonal, resp. triangular, matrix of matrices has been seen earlier, by considering the corresponding matrix $M(u)$ of linear mappings. The lower triangular (resp. upper triangular, diagonal) matrices of matrices for a given partition $(\mathbf{I}_k)$ of I form subrings of the ring of matrices $\mathbf{A}^{I \times I}$. In particular, the ring of diagonal matrices of matrices relative to the partition $(\mathbf{I}_k)$ is isomorphic to the product $\prod_{k=1}^p \operatorname{End}_\mathbf{A}(E_k)$.