Elementary transvection matrix ($I+t{E}_{ij},i\ne j$)

This matrix adds $t$ times coordinate $j$ to coordinate $i$. Left multiplication performs the corresponding row addition. Such matrices generate the special linear group over a finite field: row elimination also realizes signed interchanges and compensating diagonal scalings using products of these matrices.

Row additions reduce an invertible matrix to diagonal form without changing determinant. A determinant-one diagonal matrix is a product of diagonal two-coordinate blocks $\mathrm{diag}(a,a^{-1})$, each expressible as a product of six elementary transvection matrices. Therefore these matrices generate the special linear group over a field.

// Target: algebra.bigb