## On the matrix equation $$X+A^ TX^{-1}A=I$$.(English)Zbl 0863.15005

It is shown that the matrix equation $$(*)$$ $$X+A^TX^{-1}A=I$$ has a positive definite solution $$(X>0)$$ if and only if $$A$$ has a factorization $$A=W^TZ$$, where the matrix $$W$$ is nonsingular and the columns of $$[W^T,Z^T]^T$$ are orthonormal. In this case $$X=W^TW$$. It is also proved that equation $$(*)$$ has a solution $$X>0$$ if and only if there exist orthogonal matrices $$P$$ and $$Q$$ and diagonal matrices $$\Gamma>0$$, $$\Sigma\geq 0$$ with $$\Gamma^2+\Sigma^2=I$$ such that $$A=P^T\Gamma Q\Sigma P$$. Finally, it is shown that if $$(*)$$ has a solution $$X>0$$ then the following relations are valid $$X-AA^T>0$$, $$I-AA^T-A^TA>0$$, $$r(A)\leq 1/2$$, $$r(A+A^T)\leq 1$$, $$r(A-A^T)\leq 1$$, where $$r(A)$$ is the spectral radius of $$A$$.

### MSC:

 15A24 Matrix equations and identities

### Keywords:

matrix equation; positive definite solution; factorization
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### References:

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