Definition 5

Two matrices $X, X'$ with m rows and n columns over a ring are called equivalent if there exists an invertible square matrix $P$ of order m and an invertible square matrix $Q$ of order n such that

(40)

Clearly the relation "X and X' are equivalent" is an equivalence relation (Set Theory, II, § 6, no. 1) on the set $A^{mn}$ of matrices of type $(m, n)$ over A, which justifies the terminology.

With this definition, Proposition 6 of no. 8 can be stated by saying that when the bases are changed in two right A-modules E, F (with finite bases), the matrix of a linear mapping $u : E \to F$ with respect to the new bases is equivalent to the matrix of $u$ with respect to the old bases.

Conversely, if relation (40) holds and $u : A_d^n \to A_d^m$ is a linear mapping whose matrix is $X$ with respect to the respective canonical bases $(e_i)$ and $(f_j)$ of $A_d^n$ and $A_d^m$, then $X'$ is the matrix of $u$ with respect to the bases $(e'_i)$ and $(f'_j)$ such that $Q$ is the matrix of passage from $(e_i)$ to $(e'_i)$ and $P^{-1}$ the matrix of passage from $(f_j)$ to $(f'_j)$.

Examples of equivalent matrices. (1) Two matrices $X = (x_{ij})$ and $X' = (x'_{ij})$ with $m$ rows and $n$ columns "differ only in the order of their rows" if there exists a permutation $\sigma$ of the interval $[1, m]$ of $\mathbf{N}$, such that for every ordered pair of indices $(i, j)$, $x'_{ij} = x_{\sigma(i), j}$ (we also say that $X'$ is obtained by performing the permutation $\sigma^{-1}$ on the rows of $X$). The matrices $X$ and $X'$ are then equivalent, for $X' = PX$, where $P$ is the matrix of the permutation $\sigma^{-1}$ (cf. no. 7, Example II).

Similarly $X$ and $X'$ are said to differ only in the order of their columns if there exists a permutation $\tau$ of $[1, n]$ such that $x'_{ij} = x_{i, \tau(j)}$ for every ordered pair of indices $(i, j)$, $X$ and $X'$ are also equivalent, for $X' = XQ$ where $Q$ is the matrix of the permutation $\tau$.

Note that in the above notation $P$ is the matrix of passage from a basis $(f_j)_{1 \leq j \leq m}$ to the basis $(f_{\sigma^{-1}(j)})_{1 \leq j \leq m}$ and $Q$ the matrix of passage from a basis $(e_i)_{1 \leq i \leq n}$ to the basis $(e_{\tau(i)})_{1 \leq i \leq n}$.

(2) Let $j, k$ be distinct elements of $[1, n]$ and let $a \in A$.

Suppose that for $1 \leq i \leq m$, $x'_{ij} = x_{ij} + x_{ik}a$ and $x'_{il} = x_{il}$ for $j \neq l$ and $1 \leq i \leq m$; $X'$ is said to be derived from $X$ by adding to the column of $X$ of index $j$ the column of index $k$ multiplied on the right by $a$. In this case $X$ and $X'$ are also equivalent: for if $Q = I_n + aE_{kj}$ (an invertible triangular matrix, as seen in no. 7), then $X' = XQ$.

Similarly, let $h, i$ be two distinct elements of $[1, m]$ and $a$ an element of $A$; if $X'$ is derived from $X$ by adding to the row of $X$ of index $i$ the row of index $h$ multiplied on the left by $a$, $X$ and $X'$ are equivalent, for $X' = PX$, where $P = I_m + aE_{ih}$.

(3) Finally, if, for a given index $j$, $x'_{ij} = x_{ij}c$ for $1 \leq i \leq m$, where $c$ is invertible and $x'_{il} = x_{il}$ for $1 \leq i \leq m$ and $l \neq j$, $X$ and $X'$ are equivalent; for $X' = XQ$, where $Q$ is the matrix $\operatorname{diag}(a_k)$ with $a_j = c$, $a_k = 1$ for $k \neq j$. Then $X'$ is said to be derived from $X$ by multiplying the column of $X$ of index $j$ on the right by $a$.

Similarly, if $X'$ is derived from $X'$ by multiplying the row of $X$ of index $i$ on the left by an invertible element $c \in A$, $X'$ and $X$ are equivalent, for $X' = PX$ where $P$ is the matrix $\operatorname{diag}(b_h)$ with $b_i = c$, $b_h = 1$ for $h \neq i$.