Proposition 14

Let $X$ be a matrix of type $(m, n)$ over a field $K$. If $X$ is of rank $r$, there exist two invertible square matrices $P \in \mathbf{M}_m(K)$, $Q \in M_n(K)$ such that $P$ (resp. $Q$) is a product of matrices of order $m$ (resp. $n$) of the form $B_{ij}(\lambda)$ and

(a matrix $(\eta_{ij})$ all of whose terms are zero except the $\eta_{li}$ for $1 \leq i \leq r$, with $\eta_{li} = 1$ for $1 \leq i \leq r - 1$, $\eta_{rr} = \delta_r \neq 0$). *If $r \neq m$ or $r \neq n$, it may also be assumed that $\delta_r = 1$.

The proposition is obvious if $X = 0$; suppose therefore $X \neq 0$. If $m = n = 1$ the proposition is obvious (with $P = I_m$, $Q = I_n$, $\delta_1 \neq 0$ arbitrary). If $n = 1$, $m \geq 2$, we can apply Lemma 1 (since $X \neq 0$), which gives the desired form (58) with $r = 1$, $\delta_r = 1$. We argue by induction on $n > 1$; there exists an element $\xi_{ij} \neq 0$ in $X$; if $j = 1$, Lemma 1 can be applied and reduces the problem to the case where $X$ has the form (57). The induction hypothesis then applies to $Y$ and there are therefore invertible matrices

which are products of matrices of the form $B_{ij}(\lambda)$ of order $m - 1$ (resp. $n - 1$), such that $P'YQ'$ is of the form (58). But, if $B_{ij}(\lambda)$ belongs for example to $\mathbf{M}_{m-1}(\mathbf{K})$, then

formula (58) then follows from the formula for block products writing

If finally $j \neq 1$, it would be sufficient to consider the matrix $XB_{j1}(1)$ to reduce it to the above case.

Proposition 14 recovers Proposition 13 immediately.