Geodesic
On upper triangular nonnegative matrices
Czechoslovak Mathematical Journal, Tome 65 (2015) no. 1, pp. 1-20 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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).
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).
DOI : 10.1007/s10587-015-0158-5
Classification : 11Y05, 15A23
Keywords: upper triangular; nonnegative matrix; factorization; matrix semigroup 
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Chen, Yizhi; Zhao, Xianzhong; Liu, Zhongzhu. On upper triangular nonnegative matrices. Czechoslovak Mathematical Journal, Tome 65 (2015) no. 1, pp. 1-20. doi: 10.1007/s10587-015-0158-5

[1] Adams, D., Ardila, R., Hannasch, D., Kosh, A., McCarthy, H., Ponomarenko, V., Rosenbaum, R.: Bifurcus semigroups and rings. Involve 2 (2009), 351-356. | DOI | MR | Zbl

[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. | MR | Zbl

[3] Chuan, J. Ch., Chuan, W. F.: Factorizations in a semigroup of integral matrices. Linear Multilinear Algebra 18 (1985), 213-223. | DOI | MR | Zbl

[4] Chuan, J. Ch., Chuan, W. F.: Factorability of positive-integral matrices of prime determinants. Bull. Inst. Math., Acad. Sin. 14 (1986), 11-20. | MR | Zbl

[5] Cohn, P. M.: Noncommutative unique factorization domains. Trans. Am. Math. Soc. 109 (1963), 313-331 Errata. Ibid. 119 (1965), 552. | DOI | MR | Zbl

[6] Halava, V., Harju, T.: On Markov's undecidability theorem for integer matrices. Semigroup Forum 75 (2007), 173-180. | DOI | MR | Zbl

[7] Jacobson, B.: Matrix number theory. An example of nonunique factorization. Am. Math. Mon. 72 (1965), 399-402. | DOI | MR | Zbl

[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

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