Consider the field square matrices $M_n(\mathbb{C})$ or $M_n(\mathbb{R})$. I wish to solve the equation $AX-XA=M$ for a given $A,M$.
Obviously this is just $n^2$ linear equations and thus is trivial to actually calculate as $M_{ij}=\sum_k(A_{ik}X_{kj}-A_{kj}X_{ik})$, but I wonder if there is some closed-form equation for $X_{ij}$.
Thanks!
$\endgroup$1 Answer
$\begingroup$This equation is solved in: F. Gantmacher, Matrix theory, chapter VIII, §3.
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