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Given a finite set of matrices with integer entries, consider the question of determining whether the semigroup they generated 1) is free; 2) contains the identity matrix; 3) contains the null matrix or 4) is a group. Even for matrices of dimension $3$ , questions 1) and 3) are undecidable. For dimension $2$ , they are still open as far as we know. Here we prove that problems 2) and 4) are decidable by proving more generally that it is recursively decidable whether or not a given non singular matrix belongs...

Some decision problems on integer matrices

Christian Choffrut, Juhani Karhumäki (2010)

RAIRO - Theoretical Informatics and Applications

Given a finite set of matrices with integer entries, consider the question of determining whether the semigroup they generated 1) is free; 2) contains the identity matrix; 3) contains the null matrix or 4) is a group. Even for matrices of dimension 3, questions 1) and 3) are undecidable. For dimension 2, they are still open as far as we know. Here we prove that problems 2) and 4) are decidable by proving more generally that it is recursively decidable whether or not a given non singular matrix belongs...

Some equalities for generalized inverses of matrix sums and block circulant matrices

Yong Ge Tian (2001)

Archivum Mathematicum

Let $A_{1}, A_{2}, \dots, A_{n}$ be complex matrices of the same size. We show in this note that the Moore-Penrose inverse, the Drazin inverse and the weighted Moore-Penrose inverse of the sum $\sum_{t = 1}^{n} A_{t}$ can all be determined by the block circulant matrix generated by $A_{1}, A_{2}, \dots, A_{n}$ . In addition, some equalities are also presented for the Moore-Penrose inverse and the Drazin inverse of a quaternionic matrix.

Some Examples of Rigid Representations

Kostov, Vladimir (2000)

Serdica Mathematical Journal

*Research partially supported by INTAS grant 97-1644.Consider the Deligne-Simpson problem: give necessary and sufficient conditions for the choice of the conjugacy classes Cj ⊂ GL(n,C) (resp. cj ⊂ gl(n,C)) so that there exist irreducible (p+1)-tuples of matrices Mj ∈ Cj (resp. Aj ∈ cj) satisfying the equality M1 . . .Mp+1 = I (resp. A1+. . .+Ap+1 = 0). The matrices Mj and Aj are interpreted as monodromy operators and as matrices-residua of fuchsian systems on Riemann’s sphere. We give new examples...

Some factorizations of matrix functions in several variables

Jaromír Šimša (1992)

Archivum Mathematicum

We establish some criteria for a nonsingular square matrix depending on several parameters to be represented in the form of a matrix product of factors which depend on the single parameters.

Some Fixed Point Theorems in Boolean Algebra

Koriolan Gilezan (1980)

Publications de l'Institut Mathématique

Some Global Properties of Matrices of Functions of One Variable.

Y. SIBUYA (1965)

Mathematische Annalen

Some graphs determined by their (signless) Laplacian spectra

Muhuo Liu (2012)

Czechoslovak Mathematical Journal

Let $W_{n} = K_{1} \lor C_{n - 1}$ be the wheel graph on $n$ vertices, and let $S (n, c, k)$ be the graph on $n$ vertices obtained by attaching $n - 2 c - 2 k - 1$ pendant edges together with $k$ hanging paths of length two at vertex $v_{0}$ , where $v_{0}$ is the unique common vertex of $c$ triangles. In this paper we show that $S (n, c, k)$ ( $c \geq 1$ , $k \geq 1$ ) and $W_{n}$ are determined by their signless Laplacian spectra, respectively. Moreover, we also prove that $S (n, c, k)$ and its complement graph are determined by their Laplacian spectra, respectively, for $c \geq 0$ and $k \geq 1$ .

Some inequalities for commutators and an application to spectral variation.

R. Bhatia, F. Kittaneh, C. Davis (1991)

Aequationes mathematicae

Some inequalities for spectral variations.

Zhan, Shilin (2006)

JIPAM. Journal of Inequalities in Pure & Applied Mathematics [electronic only]

Some inequalities for sums of nonnegative definite matrices in quaternions.

Tian, Yongge, Styan, George P.H. (2005)

Journal of Inequalities and Applications [electronic only]

Some inequalities for the Khatri-Rao product of matrices.

Cao, Chong-Guang, Zhang, Xian, Yang, Zhong-Peng (2002)

ELA. The Electronic Journal of Linear Algebra [electronic only]

Some inequalities involving upper bounds for some matrix operators. I

R. Lashkaripour, D. Foroutannia (2007)

Czechoslovak Mathematical Journal

In this paper we consider the problem of finding upper bounds of certain matrix operators such as Hausdorff, Nörlund matrix, weighted mean and summability on sequence spaces $l_{p} (w)$ and Lorentz sequence spaces $d (w, p)$ , which was recently considered in [9] and [10] and similarly to [14] by Josip Pecaric, Ivan Peric and Rajko Roki. Also, this study is an extension of some works by G. Bennett on $l_{p}$ spaces, see [1] and [2].

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