Heinig, Georg Not every matrix is similar to a Toeplitz matrix. (English) Zbl 0985.15013 Linear Algebra Appl. 332-334, No. 1-3, 519-531 (2001). The inverse eigenvalue problem in matrix calculus is considered. It is known that such a problem is always solvable in a class of Toeplitz matrices and, in particular, of real symmetric Toeplitz matrices [cf. P. Delsarte and Y. Genin, Lect. Notes Control Inf. Sci. 58, 194-213 (1984; Zbl 0559.15017); H. J. Landau, J. Am. Math. Soc. 7, No. 3, 749-767 (1994; Zbl 0813.15006)]. Recently the inverse Jordan structure problem for Toeplitz matrices was formulated as follows: Is every matrix similar to a Toeplitz matrix? [cf. D. S. Mackey, N. Mackey and S. Petrovic, Linear Algebra Appl. 297, No. 1-3, 87-105 (1999; Zbl 0939.15003)]. In the paper a negative answer to this question is given, which is obtained by analyzing the structure of the intersection of kernel and range of a Toeplitz matrix. As a result, some examples of the matrices which are not similar to a Toeplitz matrix are found. Reviewer: A.A.Bogush (Minsk) Cited in 2 ReviewsCited in 5 Documents MSC: 15A29 Inverse problems in linear algebra 15A21 Canonical forms, reductions, classification 15B57 Hermitian, skew-Hermitian, and related matrices 15A18 Eigenvalues, singular values, and eigenvectors Keywords:nonsimilar matrices; inverse eigenvalue problem; symmetric Toeplitz matrices; intersection of kernel and range PDF BibTeX XML Cite \textit{G. Heinig}, Linear Algebra Appl. 332--334, 519--531 (2001; Zbl 0985.15013) Full Text: DOI References: [1] P. Delsarte, Y. Genin, Spectral properties of finite Toeplitz matrices, in: P.A. Fuhrmann (Ed.), Proceedings of the MTNS-83, Beer Sheva, Lecture Notes in Control and Information Sciences, vol. 58, Springer, New York, 1984, pp. 194-213 · Zbl 0559.15017 [2] Heinig, G., Endliche toeplitzmatrizen und zweidimensionale wiener – hopf-operatoren mit homogenem symbol, Math. nachr., 82, 29-52, (1978) · Zbl 0318.45007 [3] Heinig, G.; Jankowski, P., Kernel structure of block Hankel matrices and partial realization, Linear algebra appl., 175, 1-32, (1992) · Zbl 0756.15027 [4] Heinig, G.; Rost, K., Algebraic methods for Toeplitz-like matrices and operators, (1984), Akademie-Verlag Berlin and Birhäuser, Basel · Zbl 0549.15013 [5] Landau, H.J., The inverse eigenvalue problem for real symmetric Toeplitz matrices, J. amer. math. soc., 7, 749-767, (1994) · Zbl 0813.15006 [6] Mackey, D.S.; Mackey, N.; Petrovic, S., Is every matrix similar to a Toeplitz matrix? linear algebra appl., 297, 87-105, (1999) · Zbl 0939.15003 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.