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On the Consistency of the Matrix Equation X⊤ AX = B when B is Symmetric

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2021-04
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Springer Nature
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We provide necessary and sufficient conditions for the matrix equation X⊤AX=B to be consistent when B is a symmetric matrix, for all matrices A with a few exceptions. The matrices A, B, and X (unknown) are matrices with complex entries. We first see that we can restrict ourselves to the case where A and B are given in canonical form for congruence and, then, we address the equation with A and B in such form. The characterization strongly depends on the canonical form for congruence of A. The problem we solve is equivalent to: given a complex bilinear form (represented by A) find the maximum dimension of a subspace such that the restriction of the bilinear form to this subspace is a symmetric non-degenerate bilinear form.
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Matrix equation, Transpose, Congruence, T-Riccati equation, Canonical form for congruence, Symmetric matrix, Bilinear form
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Borobia, A., Canogar, R. & De Terán, F. (2021). On the Consistency of the Matrix Equation X⊤ AX = B when B is Symmetric. Mediterranean Journal of Mathematics, 18(2), 40.