| Title:
|
On upper triangular nonnegative matrices (English) |
| Author:
|
Chen, Yizhi |
| Author:
|
Zhao, Xianzhong |
| Author:
|
Liu, Zhongzhu |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
65 |
| Issue:
|
1 |
| Year:
|
2015 |
| Pages:
|
1-20 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We first investigate factorizations of elements of the semigroup $S$ of upper triangular matrices with nonnegative entries and nonzero determinant, provide a formula for $\rho (S)$, and, given $A\in S$, also provide formulas for $l(A)$, $L(A)$ and $\rho (A)$. As a consequence, open problem 2 and problem 4 presented in N. Baeth et al. (2011), are partly answered. Secondly, we study the semigroup of upper triangular matrices with only positive integral entries, compute some invariants of such semigroup, and also partly answer open Problem 1 and Problem 3 in N. Baeth et al. (2011). (English) |
| Keyword:
|
upper triangular |
| Keyword:
|
nonnegative matrix |
| Keyword:
|
factorization |
| Keyword:
|
matrix semigroup |
| MSC:
|
11Y05 |
| MSC:
|
15A23 |
| idZBL:
|
Zbl 06433719 |
| idMR:
|
MR3336023 |
| DOI:
|
10.1007/s10587-015-0158-5 |
| . |
| Date available:
|
2015-04-01T12:17:36Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144209 |
| . |
| Reference:
|
[1] Adams, D., Ardila, R., Hannasch, D., Kosh, A., McCarthy, H., Ponomarenko, V., Rosenbaum, R.: Bifurcus semigroups and rings.Involve 2 (2009), 351-356. Zbl 1190.20046, MR 2551131, 10.2140/involve.2009.2.351 |
| Reference:
|
[2] Baeth, N., Ponomarenko, V., Adams, D., Ardila, R., Hannasch, D., Kosh, A., McCarthy, H., Rosenbaum, R.: Number theory of matrix semigroups.Linear Algebra Appl. 434 (2011), 694-711. Zbl 1250.11104, MR 2746077 |
| Reference:
|
[3] Chuan, J. Ch., Chuan, W. F.: Factorizations in a semigroup of integral matrices.Linear Multilinear Algebra 18 (1985), 213-223. Zbl 0593.15006, MR 0828404, 10.1080/03081088508817688 |
| Reference:
|
[4] Chuan, J. Ch., Chuan, W. F.: Factorability of positive-integral matrices of prime determinants.Bull. Inst. Math., Acad. Sin. 14 (1986), 11-20. Zbl 0593.15007, MR 0861146 |
| Reference:
|
[5] Cohn, P. M.: Noncommutative unique factorization domains.Trans. Am. Math. Soc. 109 (1963), 313-331 Errata. Ibid. 119 (1965), 552. Zbl 0136.31203, MR 0155851, 10.1090/S0002-9947-1963-0155851-X |
| Reference:
|
[6] Halava, V., Harju, T.: On Markov's undecidability theorem for integer matrices.Semigroup Forum 75 (2007), 173-180. Zbl 1131.20042, MR 2351930, 10.1007/s00233-007-0714-x |
| Reference:
|
[7] Jacobson, B.: Matrix number theory. An example of nonunique factorization.Am. Math. Mon. 72 (1965), 399-402. Zbl 0134.25204, MR 1533217, 10.2307/2313504 |
| Reference:
|
[8] Jacobson, B., Wisner, R. J.: Matrix number theory. I: Factorization of $2 \times 2$ unimodular matrices.Publ. Math. Debrecen 13 (1966), 67-72. MR 0202696 |
| . |
