MATH319 Slides 125 Sylvester’s equation 127 Sylvester’s criterion

126 Cases

Proof. In terms of the matrix entries, we can write $A=[a_{jk}],$ $B=[b_{jk}]$ and $C=[c_{jk}]$ , and then $Y=[y_{jk}]$ is given by the system

\sum_{\ell=1}^{n}a_{j\ell}y_{\ell k}+\sum_{\ell=1}^{n}y_{j\ell}b_{\ell k}=-c_{% jk},

which is a linear system of $n^{2}$ equations in the $n^{2}$ unknowns $[y_{jk}]$ . So the above possibilities arise from the general theory of linear equations. Gauss–Jordan elimination reduces this system to reduced echelon form and gives the solutions in cases (i) and (ii). In case (iii), the system is inconsistent.