Czechoslovak Mathematical Journal

, Volume 58, Issue 4, pp 1185–1193

# Noncirculant Toeplitz matrices all of whose powers are Toeplitz

• Kent Griffin
• Jeffrey L. Stuart
• Michael J. Tsatsomeros
Article

## Abstract

Let a, b and c be fixed complex numbers. Let M n (a, b, c) be the n × n Toeplitz matrix all of whose entries above the diagonal are a, all of whose entries below the diagonal are b, and all of whose entries on the diagonal are c. For 1 ⩽ kn, each k × k principal minor of M n (a, b, c) has the same value. We find explicit and recursive formulae for the principal minors and the characteristic polynomial of M n (a, b, c). We also show that all complex polynomials in M n (a, b, c) are Toeplitz matrices. In particular, the inverse of M n (a, b, c) is a Toeplitz matrix when it exists.

## Keywords

Toeplitz matrix Toeplitz inverse Toeplitz powers principal minor Fibonacci sequence

## References

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K. Griffin, M. J. Tsatsomeros: Principal minors, Part II: The principal minor assignment problem. Linear Algebra Appl. 419 (2006), 125–171.
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N. M. Huang, R. E. Cline: Inversion of persymmetric matrices having Toeplitz inverses. J. Assoc. Comput. Mach. 19 (1972), 437–444.
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T. Shalom: On algebras of Toeplitz matrices. Linear Algebra Appl. 96 (1987), 211–226.

## Authors and Affiliations

• Kent Griffin
• 1
• Jeffrey L. Stuart
• 2
• Michael J. Tsatsomeros
• 3
1. 1.Santa MonicaUSA
2. 2.Mathematics DepartmentPacific Lutheran UniversityTacomaUSA
3. 3.Mathematics DepartmentWashington State UniversityPullmanUSA