Abstract. By applying the matrix rank method, the set of symmetric matrix solutions with prescribed rank to the matrix equation AX = B is found. An expression is provided for the optimal approximation to the set of the minimal rank solutions.

Abstract. In this paper, the following problems are discussed. Problem I. Given matrices A ∈ R n×m and D ∈ R m×m , find X ∈ ASR n P such that A T XA = D, whereProblem II. Given a matrixX ∈ R n×n , findX ∈ S E such thatwhere · is the Frobenius norm, and S E is the solution set of Problem I. Expressions for the general solution of Problem I are derived. Necessary and sufficient conditions for the solvability of Problem I are provided. For Problem II, an expression for the solution is given as well.

Let n n complex matrices P and Q be nontrivial generalized reflection matrices, i.e.

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