Hermitian solutions of the equation \(X=Q+NX^{-1}N^*\). (English) Zbl 0876.15011

The authors consider the set \({\mathcal X}\) of Hermitian matrices \(X\) which satisfy \(X = Q + NX^{-1}N^*\) where \(Q\) and \(N\) are specified \(n \times n\) matrices over \({\mathbb C}\) and \(Q\) is Hermitian positive definite. They show that that there is a one-to-one correspondence between the elements \(X\) in \({\mathcal X}\) and the factorizations of the matrix polynomial \(N(s,t) := Qst + Ns^{2} - N^*t^{2}\) in the form \(N(s,t) = (tI + sM^*)X(sI - tM)\) for some \(M\). They also show that under the usual ordering for Hermitian matrices \({\mathcal X}\) has a unique maximal element \(X_{+}\) and, if \(N\) is nonsingular, a unique minimal element \(X_{-}\). Moreover, \(X_{+}\) can be computed as the limit of the sequence \(\{X_{k}\}\) where \(X_{0} := Q\) and \(X_{k+1} := Q + NX^{-1}_{k}N^*\).
Reviewer: J.D.Dixon (Ottawa)


15A24 Matrix equations and identities
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