8 results
Search Results
2. Permutation-like matrix groups with a maximal cycle of length power of two.
- Author
-
Deng, Guodong and Fan, Yun
- Subjects
- *
PERMUTATION groups , *MATRIX groups , *NUMBER theory , *GROUP theory , *MATHEMATICAL analysis - Abstract
If every element of a matrix group is similar to a permutation matrix, then it is called a permutation-like matrix group. References [3] , [4] and [5] showed that, if a permutation-like matrix group contains a maximal cycle such that the maximal cycle generates a normal subgroup and the length of the maximal cycle equals a prime, or a square of a prime, or a power of an odd prime, then the permutation-like matrix group is similar to a permutation matrix group. In this paper, we prove that if a permutation-like matrix group contains a maximal cycle such that the maximal cycle generates a normal subgroup and the length of the maximal cycle equals any power of 2, then it is similar to a permutation matrix group. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
3. Lengths of quasi-commutative pairs of matrices.
- Author
-
Guterman, A.E., Markova, O.V., and Mehrmann, V.
- Subjects
- *
COMMUTATIVE algebra , *MATRICES (Mathematics) , *SET theory , *NUMBER theory , *LINEAR statistical models - Abstract
In this paper we discuss some partial solutions of the length conjecture which describes the length of a generating system for matrix algebras. We consider mainly the sets of two matrices which are quasi-commuting. It is shown that in this case the length function is linearly bounded. We also analyze which particular natural numbers can be realized as the lengths of certain special generating sets and prove that for commuting or product-nilpotent pairs all possible numbers are realizable, however there are non-realizable values between lower and upper bounds for the other quasi-commuting pairs. In conclusion we also present several related open problems. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
4. The effect of assuming the identity as a generator on the length of the matrix algebra.
- Author
-
Laffey, Thomas, Markova, Olga, and Šmigoc, Helena
- Subjects
- *
MATRICES (Mathematics) , *SET theory , *VECTOR spaces , *FINITE mixture models (Statistics) , *MATHEMATICAL functions - Abstract
Let M n ( F ) be the algebra of n × n matrices and let S be a generating set of M n ( F ) as an F -algebra. The length of a finite generating set S of M n ( F ) is the smallest number k such that words of length not greater than k generate M n ( F ) as a vector space. Traditionally the identity matrix is assumed to be automatically included in all generating sets S and counted as a word of length 0. In this paper we discuss how the problem changes if this assumption is removed. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
5. Q-less QR decomposition in inner product spaces.
- Author
-
Fan, H.-Y., Zhang, L., Chu, E.K.-w., and Wei, Y.
- Subjects
- *
INNER product , *MATHEMATICS , *NUMERICAL analysis , *EQUATIONS , *ALGEBRA - Abstract
Tensor computation is intensive and difficult. Invariably, a vital component is the truncation of tensors, so as to control the memory and associated computational requirements. Various tensor toolboxes have been designed for such a purpose, in addition to transforming tensors between different formats. In this paper, we propose a simple Q-less QR truncation technique for tensors { x ( i ) } with x ( i ) ∈ R n 1 × ⋯ × n d in the simple and natural Kronecker product form. It generalizes the QR decomposition with column pivoting, adapting the well-known Gram–Schmidt orthogonalization process. The main difficulty lies in the fact that linear combinations of tensors cannot be computed or stored explicitly. All computations have to be performed on the coefficients α i in an arbitrary tensor v = ∑ i α i x ( i ) . The orthonormal Q factor in the QR decomposition X ≡ [ x ( 1 ) , ⋯ , x ( p ) ] = Q R cannot be computed but expressed as X R − 1 when required. The resulting algorithm has an O ( p 2 d n ) computational complexity, with n = max n i . Some illustrative examples in the numerical solution of tensor linear equations are presented. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
6. Permutation-like matrix groups with a maximal cycle of power of odd prime length.
- Author
-
Deng, Guodong and Fan, Yun
- Subjects
- *
PERMUTATION groups , *MATRIX groups , *MAXIMAL functions , *MATHEMATICAL proofs , *MATHEMATICAL analysis - Abstract
If every element of a matrix group is similar to a permutation matrix, then it is called a permutation-like matrix group. Refs. [3] and [4] showed that, if a permutation-like matrix group contains a maximal cycle of length equal to a prime or a square of a prime and the maximal cycle generates a normal subgroup, then it is similar to a permutation matrix group. In this paper, we prove that if a permutation-like matrix group contains a maximal cycle of length equal to any power of any odd prime and the maximal cycle generates a normal subgroup, then it is similar to a permutation matrix group. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
7. The product of matrix subspaces.
- Author
-
Huhtanen, Marko
- Subjects
- *
SUBSPACES (Mathematics) , *MATRICES (Mathematics) , *FACTORIZATION , *PERFORMANCE evaluation , *GROUP theory - Abstract
In factoring matrices into the product of two matrices, operations are typically performed with elements restricted to matrix subspaces. Such modest structural assumptions are realistic, for example, in large scale computations. This paper is concerned with analyzing associated matrix geometries. Curvature of the product of two matrix subspaces is assessed. The case of vanishing curvature gives rise to the notion of factorizable matrix subspace. This can be regarded as an analogue of the internal Zappa–Szép product in group theory. Interpreted in this way, several classical instances are encompassed by this structure. The Craig–Sakamoto theorem fits naturally into this framework. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
8. Tridiagonal matrices with nonnegative entries
- Author
-
Nomura, Kazumasa and Terwilliger, Paul
- Subjects
- *
MATRICES (Mathematics) , *PERMUTATIONS , *IDEMPOTENTS , *MULTIPLICITY (Mathematics) , *GRAPH theory , *LINEAR algebra , *SET theory - Abstract
Abstract: In this paper, we characterize the nonnegative irreducible tridiagonal matrices and their permutations, using certain entries in their primitive idempotents. Our main result is summarized as follows. Let d denote a nonnegative integer. Let A denote a matrix in and let denote the roots of the characteristic polynomial of A. We say A is multiplicity-free whenever these roots are mutually distinct and contained in . In this case will denote the primitive idempotent of A associated with . We say A is symmetrizable whenever there exists an invertible diagonal matrix such that is symmetric. Let denote the directed graph with vertex set , where whenever and . Theorem. Assume that each entry of is nonnegative. Then the following are equivalent for . [(i)] The graph is a bidirected path with endpoints , : [(ii)] The matrix is symmetrizable and multiplicity-free. Moreover the -entry of times is independent of i for , and this common value is nonzero. Recently Kurihara and Nozaki obtained a theorem that characterizes the -polynomial property for symmetric association schemes. We view the above result as a linear algebraic generalization of their theorem. [Copyright &y& Elsevier]
- Published
- 2011
- Full Text
- View/download PDF
Discovery Service for Jio Institute Digital Library
For full access to our library's resources, please sign in.