Noncirculant Toeplitz matrices all of whose powers are Toeplitz |
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| Authors: | Kent Griffin Jeffrey L Stuart Michael J Tsatsomeros |
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| Institution: | (1) 1336 Princeton St., Santa Monica, CA 90404, USA;(2) Mathematics Department, Pacific Lutheran University, Tacoma, WA 98447, USA;(3) Mathematics Department, Washington State University, Pullman, WA 99164, USA |
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| 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 ⩽ k ⩽ n, 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. |
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| Keywords: | Toeplitz matrix Toeplitz inverse Toeplitz powers principal minor Fibonacci sequence |
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