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Hermitian solutions of the equation $X=Q+N{X}^{-1}{N}^{*}$. (English) Zbl 0876.15011
The authors consider the set $𝒳$ of Hermitian matrices $X$ which satisfy $X=Q+N{X}^{-1}{N}^{*}$ where $Q$ and $N$ are specified $n×n$ matrices over $ℂ$ and $Q$ is Hermitian positive definite. They show that that there is a one-to-one correspondence between the elements $X$ in $𝒳$ and the factorizations of the matrix polynomial $N\left(s,t\right):=Qst+N{s}^{2}-{N}^{*}{t}^{2}$ in the form $N\left(s,t\right)=\left(tI+s{M}^{*}\right)X\left(sI-tM\right)$ for some $M$. They also show that under the usual ordering for Hermitian matrices $𝒳$ 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 $\left\{{X}_{k}\right\}$ where ${X}_{0}:=Q$ and ${X}_{k+1}:=Q+N{X}_{k}^{-1}{N}^{*}$.

##### MSC:
 15A24 Matrix equations and identities
##### Keywords:
matrix equation; Hermitian matrices; matrix polynomial