On the matrix equation \(X^{-1}X^*=A\) (Q1810102)

The author considers the matrix equation \((*)\) \(X^{-1}X^*=A\) which is equivalent to the homogeneous discrete Lyapunov equation \(A^*XA=X\) and to \(XA-A^{-1*}X=0\) which is a particular case of the homogeneous continuous Sylvester equation \(XA-BX=0\) (notice that by \((*)\) the matrix \(A\) must be non-degenerate). He shows that when equation \((*)\) is solvable, then for each eigenvalue \(\lambda\) of \(A\) (\(|\lambda |\neq 1\)), \(1/\bar{\lambda}\) is also an eigenvalue of \(A\), with the same number and sizes of Jordan blocks.