A matrix A ∈ C n×n is unitarily quasidiagonalizable if A can be brought by a unitary similarity transformation to a block diagonal form with 1 × 1 and 2 × 2 diagonal blocks. In particular, the square roots of normal matrices, i.e., the so-called quadratically normal matrices are unitarily quasidiagonalizable. A matrix A ∈ C n×n is congruence-normal if \( B = A\overline A \) is a conventional normal matrix. We show that every congruence-normal matrix A can be brought by a unitary congruence transformation to a block diagonal form with 1 × 1 and 2 × 2 diagonal blocks. Our proof emphasizes andexploitsalikenessbetween theequations X 2 = B and \( X\overline X = B \) for a normal matrix B. Bibliography: 13 titles.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. Grone, C. R. Johnson, E. M. Sa, and H. Wolkowicz, “Normal matrices,” Linear Algebra Appl., 87, 213–225 (1987).
L. Elsner and Kh. D. Ikramov, “Normal matrices: an update,” Linear Algebra Appl., 285, 291–303 (1998).
H. Faßbender and Kh. D. Ikramov, “Conjugate-normal matrices: a survey,” Linear Algebra Appl., 429, 1425–1441 (2008).
M. Vujičić, F. Herbut, and G. Vujičić, “Canonical form for matrices under unitary congruence transformations. I. Conjugate-normal matrices,” SIAM J. Appl. Math., 23, 225–238 (1972).
H. Faßbender and Kh. D. Ikramov, “Some observations on the Youla form and conjugate-normal matries,” Linear Algebra Appl., 422, 29–38 (2007).
F. Herbut, P. Lonke, and M. Vujičić, “Canonial form for matrices under unitary congruence transformations. II. Congruene-normal matrices,” SIAM J. Appl. Math., 26, 794–805 (1974).
F. Kittaneh, “On the structure of polynomially normal operators,” Bull. Austral. Math. Soc., 30, 11–18 (1984).
H. Radjavi and P. Rosenthal, “On roots of normal operators,” J. Math. Anal. Appl., 34, 653–664 (1971).
T. J. Laffey, “A normality criterion for an algebra of matrices,” Linear Algebra Appl., 25, 169–174 (1979).
R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge Univ. Press, Cambridge (1985).
R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge Univ. Press, Cambridge (1991).
R. A. Horn and D. I. Merino, “A real-coninvolutory analog of the polar decomposition,” Linear Algebra Appl., 190, 209–227 (1993).
M. N. M. Abara, D. I. Merino, and A. T. Paras, “Skew-coninvolutory matrices,” Linear Algebra Appl., 426, 540–557 (2007).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 367, 2009, pp. 45–66.
Rights and permissions
About this article
Cite this article
Ikramov, K.D., Fassbender, H. Quadratically normal and congruence-normal matrices. J Math Sci 165, 521–532 (2010). https://doi.org/10.1007/s10958-010-9822-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-010-9822-3