Let a complex matrix A be the direct sum of its square submatrices B and C that have no common eigenvalues. Then every matrix X belonging to the centralizer of A has the same block diagonal form as the matrix A. This paper discusses how the conditions on the submatrices B and C should be modified for an analogous assertion about the congruent centralizer of A, which is the set of matrices X such that X * AX = A, to be valid. Also the question whether the matrices in the congruent centralizer are block diagonal if A is a block anti-diagonal matrix is considered. Bibliography: 2 titles.
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R. A. Horn and C. R. Johnson, Matrix Analysis. Second edition, Cambridge University Press, Cambridge (2013).
D. Kressner, C. Schröder, and D. S. Watkins, “Implicit QR algorithms for palindromic and even eigenvalue problems,” Numer. Algorithms, 51, 209–238 (2009).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 453, 2016, pp. 96–103.
Translated by Kh. D. Ikramov.
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Ikramov, K.D. The Congruent Centralizer of a Block Diagonal Matrix. J Math Sci 224, 877–882 (2017). https://doi.org/10.1007/s10958-017-3457-6
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DOI: https://doi.org/10.1007/s10958-017-3457-6