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Hermitian solutions of the equation $X=Q+NX\sp{-1}N\sp*$. (English) Zbl 0876.15011
The authors consider the set ${\cal X}$ of Hermitian matrices $X$ which satisfy $X = Q + NX^{-1}N^*$ where $Q$ and $N$ are specified $n \times n$ matrices over ${\Bbb C}$ and $Q$ is Hermitian positive definite. They show that that there is a one-to-one correspondence between the elements $X$ in ${\cal X}$ 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 ${\cal X}$ 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 $X_{k+1} := Q + NX^{-1}_{k}N^*$.

MSC:
15A24Matrix equations and identities
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