Previous |  Up |  Next


Toeplitz matrix; Toeplitz inverse; Toeplitz powers; principal minor; Fibonacci sequence
Let $a$, $b$ and $c$ be fixed complex numbers. Let $M_n(a,b,c)$ be the $n\times n$ 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 $1\leq k\leq n$, each $k\times k$ 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.
[1] Griffin, K.: Solving the principal minor assignment problem and related computations. PhD. Dissertation Washington State University Washington (2006). MR 2709319
[2] Griffin, K., Tsatsomeros, M. J.: Principal minors, Part I: A method for computing all the principal minors of a matrix. Linear Algebra Appl. 419 (2006), 107-124. MR 2263114
[3] Griffin, K., Tsatsomeros, M. J.: Principal minors, Part II: The principal minor assignment problem. Linear Algebra Appl. 419 (2006), 125-171. MR 2263115
[4] Huang, N. M., Cline, R. E.: Inversion of persymmetric matrices having Toeplitz inverses. J. Assoc. Comput. Mach. 19 (1972), 437-444. DOI 10.1145/321707.321714 | MR 0312704 | Zbl 0259.65032
[5] Shalom, T.: On algebras of Toeplitz matrices. Linear Algebra Appl. 96 (1987), 211-226. DOI 10.1016/0024-3795(87)90345-4 | MR 0910995 | Zbl 0644.15005
Partner of
EuDML logo