| Title:
|
Noncirculant Toeplitz matrices all of whose powers are Toeplitz (English) |
| Author:
|
Griffin, Kent |
| Author:
|
Stuart, Jeffrey L. |
| Author:
|
Tsatsomeros, Michael J. |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
58 |
| Issue:
|
4 |
| Year:
|
2008 |
| Pages:
|
1185-1193 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
Toeplitz matrix |
| Keyword:
|
Toeplitz inverse |
| Keyword:
|
Toeplitz powers |
| Keyword:
|
principal minor |
| Keyword:
|
Fibonacci sequence |
| MSC:
|
11B37 |
| MSC:
|
11B39 |
| MSC:
|
15A15 |
| MSC:
|
15A57 |
| idZBL:
|
Zbl 1174.15011 |
| idMR:
|
MR2471175 |
| . |
| Date available:
|
2010-07-21T08:14:38Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/140449 |
| . |
| Reference:
|
[1] Griffin, K.: Solving the principal minor assignment problem and related computations.PhD. Dissertation Washington State University Washington (2006). MR 2709319 |
| Reference:
|
[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 |
| Reference:
|
[3] Griffin, K., Tsatsomeros, M. J.: Principal minors, Part II: The principal minor assignment problem.Linear Algebra Appl. 419 (2006), 125-171. MR 2263115 |
| Reference:
|
[4] Huang, N. M., Cline, R. E.: Inversion of persymmetric matrices having Toeplitz inverses.J. Assoc. Comput. Mach. 19 (1972), 437-444. Zbl 0259.65032, MR 0312704, 10.1145/321707.321714 |
| Reference:
|
[5] Shalom, T.: On algebras of Toeplitz matrices.Linear Algebra Appl. 96 (1987), 211-226. Zbl 0644.15005, MR 0910995, 10.1016/0024-3795(87)90345-4 |
| . |
