## 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)

### MSC:

 15A24 Matrix equations and identities

### Keywords:

matrix equation; Hermitian matrices; matrix polynomial
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### References:

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