## Some properties for the existence of a positive definite solution of matrix equation $$X+A^*X^{-2^m}A=I$$.(English)Zbl 1031.15015

Let $$P(n)$$ be the set of all positive definite $$(n\times n)$$ matrices; $$I$$ is the $$(n\times n)$$ identity matrix; and $$m, n$$ $$\in \mathbb{N}$$. The authors studies the matrix equation $X+A^*X^{-2^m}A=I \tag{1}$ with given matrix $$A\in P(n)$$ and unknown matrix $$X\in P(n)$$. They give necessary and sufficient conditions for the solvability of equation (1). The main results of this paper are:
Theorem 3. If there exist numbers $$a, b$$ satisfying $$0<a<b<1$$ and the inequalities $a^{2m}(1-a)I<AA^*<b^{2m}(1-b)I$ hold, then (1) has a positive definite solution.
Theorem 6. If (1) has a positive definite solution $$X$$, then $A^*A+ (AA^*)^{\frac{1}{2m}}<I.$

### MSC:

 15A24 Matrix equations and identities 15B48 Positive matrices and their generalizations; cones of matrices

### Keywords:

matrix equation; positive definite solution
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### References:

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