Abstract
In studying the reduction of a complex n × n matrix A to its Hessenberg form by the Arnoldi algorithm, T. Huckle discovered that an irreducible Hessenberg normal matrix with a normal leading principal m × m submatrix, where 1 < m < n, actually is tridiagonal. We prove a similar assertion for the conjugate-normal matrices, which play the same role in the theory of unitary congruences as the conventional normal matrices in the theory of unitary similarities. This fact is stated as a purely matrix-theoretic theorem, without any reference to Arnoldi-like algorithms. Bibliography: 2 titles.
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T. Huckle, “The Arnoldi method for normal matrices,” SIAM J. Matrix Anal. Appl., 15, No. 2, 479–489 (1994).
L. Elsner and Kh. D. Ikramov, “On normal matrices with normal principles submatrices,” Zap. Nauchn. Semin. POMI, 229, 63–94 (1995).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 346, 2007, pp. 21–25.
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Ghasemi Kamalvand, M., Ikramov, K.D. Conjugate-normal matrices with conjugate-normal submatrices. J Math Sci 150, 1926–1928 (2008). https://doi.org/10.1007/s10958-008-0106-0
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DOI: https://doi.org/10.1007/s10958-008-0106-0