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Acta Mathematica Academiae Paedagogicae Nyíregyháziensis, Vol. 21, No. 1, pp. 33-42 (2005)
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# The general Hermitian nonnegative-definite solution to the matrix equation $AXA^{\ast }+BYB^{\ast }=C$

## Xian Zhang

Heilongjiang University

**Abstract:**
A matrix pair $\left( X_{0},Y_{0}\right) $ is called a Hermitian nonnegative-definite solution to the matrix equation if $X_{0}$ and $Y_{0}$ are Hermitian nonnegative-definite and satisfy $AX_{0}A^{\ast }+BY_{0}B^{\ast }=C$. We give necessary and sufficient conditions for the existence of a Hermitian nonnegative-definite solution to the matrix equation, and further derive a representation of the general Hermitian nonnegative-definite solution to the equation when it has such solutions. An example shows these advantages of the proposed approach.

**Keywords:** Hermitian nonnegative-definite solution, matrix equation, generalized inverse, singular value decomposition.

**Classification (MSC2000):** 15A06; 19A09

**Full text of the article:**

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© 2005 ELibM and
FIZ Karlsruhe / Zentralblatt MATH
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