Abstract
We provide necessary and sufficient conditions for the matrix equation \(X^\top A X=B\) 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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Acknowledgements
We thank an anonymous referee for several comments that allowed us to improve the original version. This work has been partially supported by the Ministerio de Economía y Competitividad of Spain through Grant MTM2015-65798-P (Secretaría de Estado de Investigación, Desarrollo e Innovación) (F. De Terán), and by the Ministerio de Ciencia, Innovación y Universidades of Spain through Grant MTM2017–90682–REDT.
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Borobia, A., Canogar, R. & De Terán, F. On the Consistency of the Matrix Equation \(X^\top A X=B\) when B is Symmetric. Mediterr. J. Math. 18, 40 (2021). https://doi.org/10.1007/s00009-020-01656-7
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DOI: https://doi.org/10.1007/s00009-020-01656-7
Keywords
- Matrix equation
- transpose
- congruence
- \(\top \)-Riccati equation
- canonical form for congruence
- symmetric matrix
- bilinear form