Massimiliano Fasi: Solving rational matrix equations

Abstract:   We consider rational matrix equations of the form \(p(X) q(X)^{-1} = A\), where \(A\) is a complex square matrix and \(p\) and \(q\) are polynomials. It is easy to see that any solution \(X\) also satisfies \(p(X) = A\;q(X)\). As it turns out, the other implication is also true, and we can focus our attention on the latter simpler equation.

We develop a novel Schur method for the computation of primary solutions of the aforementioned equations, which generalises the algorithm of Smith [SIAM. J. Matrix Anal. & Appl., 24 (2003), pp. 971–989] for the computation of primary \(p\)th roots, and has a similar computational cost. We show that the algorithm can be implemented using only real arithmetic if \(A\) is real, and discuss its applications to the computation of matrix functions.