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BY 4.0 license Open Access Published by De Gruyter Open Access September 24, 2019

A note on multilevel Toeplitz matrices

  • Lei Cao and Selcuk Koyuncu EMAIL logo
From the journal Special Matrices


Chien, Liu, Nakazato and Tam proved that all n × n classical Toeplitz matrices (one-level Toeplitz matrices) are unitarily similar to complex symmetric matrices via two types of unitary matrices and the type of the unitary matrices only depends on the parity of n. In this paper we extend their result to multilevel Toeplitz matrices that any multilevel Toeplitz matrix is unitarily similar to a complex symmetric matrix. We provide a method to construct the unitary matrices that uniformly turn any multilevel Toeplitz matrix to a complex symmetric matrix by taking tensor products of these two types of unitary matrices for one-level Toeplitz matrices according to the parity of each level of the multilevel Toeplitz matrices. In addition, we introduce a class of complex symmetric matrices that are unitarily similar to some p-level Toeplitz matrices.


[1] Levon Balayan and Stephan Ramon Garcia. Unitary equivalence to a complex symmetric matrix: geometric criteria. Oper. Matrices, 4(1):53–76, 2010.10.7153/oam-04-02Search in Google Scholar

[2] Mao-Ting Chien, Jianzhen Liu, Hiroshi Nakazato, and Tin-Yau Tam. Toeplitz matrices are unitarily similar to symmetric matrices. Linear Multilinear Algebra, 65(10):2131–2144, 2017.10.1080/03081087.2017.1330865Search in Google Scholar

[3] Stephan Ramon Garcia, Daniel E. Poore, and Madeline K. Wyse. Unitary equivalence to a complex symmetric matrix: a modulus criterion. Oper. Matrices, 5(2):273–287, 2011.10.7153/oam-05-19Search in Google Scholar

[4] Stephan Ramon Garcia and Mihai Putinar. Complex symmetric operators and applications. II. Trans. Amer. Math. Soc., 359(8):3913–3931, 2007.10.1090/S0002-9947-07-04213-4Search in Google Scholar

[5] Roger A. Horn and Charles R. Johnson. Matrix analysis. Cambridge University Press, Cambridge, second edition, 2013.Search in Google Scholar

[6] E. E. Tyrtyshnikov and N. L. Zamarashkin. Spectra of multilevel Toeplitz matrices: advanced theory via simple matrix relationships. Linear Algebra Appl., 270:15–27, 1998.10.1016/S0024-3795(97)80001-8Search in Google Scholar

Received: 2019-08-07
Accepted: 2019-09-12
Published Online: 2019-09-24

© 2019 Lei Cao et al., published by De Gruyter Open

This work is licensed under the Creative Commons Attribution 4.0 International License.

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