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Not every matrix is similar to a Toeplitz matrix. (English) Zbl 0985.15013
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)

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
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[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
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