Monomial Matrices Research Articles

On the Homology of GLnand Higher Pre-Bloch Groups

AbstractFor every integern> 1 and infinite fieldFwe construct a spectral sequence converging to the homology of GLn(F) relative to the group of monomial matrices GMn(F). Some entries inE2-terms of these spectral sequences may be interpreted as a natural generalization of the Bloch group to higher dimensions. These groups may be characterized as homology of GLnrelatively to GLn-1and GMn. We apply the machinery developed to the investigation of stabilization maps in homology of General Linear Groups.

The Finite Irreducible Monomial Linear Groups of Degree 4

This paper contains an irredundant listing of the finite irreducible monomial subgroups of GL(4,C). The groups are listed up to conjugacy and are given explicitly by generating sets of monomial matrices.

Isometries for the induced c-norm on square matrices and some related results

Let F n be the linear space of column vectors with n coordinates over F = R or C. Denote by GP( n) the group of n × n generalized permutation matrices, i.e., matrices with exactly one nonzero entry with magnitude 1 in each row and column. Given a nonzero vector c = ( c 1 …, c n ) t ∈ F n , define the c-norm of x = ( x 1,…, x n ) t ∈ F n by | x | c = max ⁡ { | c t P x | : P ∈ GP ( n ) } , and the induced c-norm of A ∈ F n × n by | | A | | c = max ⁡ { | A y | c : | y | c ⩽ 1 } . We characterize, the isometries for induced c-norms. To achieve our goal, we study the geometric properties of F n × n, and obtain some inequalities related to the induced c-norm as by-products. These results are of independent interest.

Isometries for the induced c-Norm on square matrices and some related results

Let F n be the linear space of column vectors with n coordinates over F = R or C . Denote by GP( n ) the group of n × n generalized permutation matrices, i.e., matrices with exactly one nonzero entry with magnitude 1 in each row and column. Given a nonzero vector c = ( c 1 …, c n ) t ∈ F n , define the c -norm of x = ( x 1 ,…, x n ) t ∈ F n by | x | c = max ⁡ { | c t P x | : P ∈ GP ( n ) } , and the induced c -norm of A ∈ F n × n by | | A | | c = max ⁡ { | A y | c : | y | c ⩽ 1 } . We characterize, the isometries for induced c -norms. To achieve our goal, we study the geometric properties of F n × n , and obtain some inequalities related to the induced c -norm as by-products. These results are of independent interest.

On SL2-actions of complexity one

This paper is devoted to the actions of the group SL2 with two-dimensional generic orbit on normal three-dimensional affine algebraic varieties. We obtain a complete classification of such actions whenever the stabilizer of general position is the subgroup of monomial matrices. We also prove some results for arbitrary actions of complexity one of reductive groups.

Cumulant-based blind identification of linear multi-input-multi-output systems driven by colored inputs

The blind identification problem of a linear multi-input-multi-output (MIMO) system is widely noticed by many researchers in diverse fields due to its relevance to blind signal separation. However, such a problem is ill-posed and has no unique solution. Therefore, we can only find a solution of the problem within an equivalence class. We address the blind identification problem of the linear MIMO system driven by unobservable colored inputs using higher order statistics (HOS), particularly the fourth-order cumulants, of the outputs, where the unobservable inputs are mutually independent but temporally colored linear processes. We first define the set, which is denoted by S, of stable scalar transfer functions and then define the notion of a generalized permutation matrix (which is abbreviated by a g-matrix) over S. Then, it is shown that the transfer function matrix of an unknown system is identified only up to post-multiplication by a g matrix. This result is applied to identifying FIR systems for blind signal separation.

Identification of multichannel MA parameters using higher-order statistics

The identification of multichannel moving average (MA) parameter matrices { H ( k)} using fourth-order output cumulants is considered. By analyzing the eigenstructures of the cumulant matrices, it is shown that the MA parameters matrices can be identified uniquely up to a post-multiplication of monomial matrices if H (0) does not have columns that are pairwise colinear and H = [ H T(0), …, H T( L)] T has full column rank. The constructive proof of this condition suggests a possible closed-form identification algorithm.

A Characterization of Positive Constrictive Operators

In this note we prove that if T is a positive operator on a real Banach lattice, then T is constrictive if and only if that T has the operator matrix decomposition T(T1 0) -0 T2J where T1 is a power-bounded generalized permutation matrix on a finitedimensional space and T2n -* 0 strongly.

A characterization of positive constrictive operators

In this note we prove that if T is a positive operator on a real Banach lattice, then T is constrictive if and only if that T has the operator matrix decomposition \[ T = ( T 1 a m p ; 0 0 a m p ; T 2 ) , T = \left ( {\begin {array}{*{20}{c}} {{T_1}} & 0 \\ 0 & {{T_2}} \\ \end {array} } \right ), \] where T 1 {T_1} is a power-bounded generalized permutation matrix on a finite-dimensional space and T 2 n → 0 T_2^n \to 0 strongly.

The automorphism group of Generalized Reed-Muller codes

We prove that the automorphism group of Generalized Reed-Muller codes is the general linear nonhomogeneous group. The Generalized Reed-Muller codes are introduced by Kasami, Lin and Peterson. An extensive study was made by Delsarte, Goethals and Mac-Williams; our result follows their description of the minimum weight codewords. An automorphism of a cyclic q-ary code is here a substitution over the field GF( q m ). In the more general case where the automorphisms are defined by monomial matrices, we also obtain the automorphism group (called the monomial group) as the direct product of the general linear nonhomogeneous group with the multiplicative group of the alphabet field.

Certain isometries on [formula omitted

Let R n be the linear space of all real column vectors with n coordinates. Given x=(x 1,…,x n) t ∈ R n , its ( p, k)-norm is defined by ‖x‖ p, k = max (|x i 1| p+⋯+|x i k| p) 1⧸p:1 ⩽ i 1 <⋯< i k ⩽ n where k is a positive integer satisfying 1 ⩽ k ⩽ n and p satisfies 1 ⩽ p ⩽ ∞, whereas its c-norm is defined by |x| c = max{c tPx: P is a generalized permutation matrix}, for any nonzero c ∈ R n . This paper characterizes the isometries for c-norms and ( p, k)-norms. Some related results are also considered.

Schur extensions of monomial matrix groups

For each Schur function, or generalized matrix function, corresponding to a linear character of a subgroup of the symmetric group, we construct a matrix semigroup containing the monomial group defined by the subgroup. The matrix semigroup constructed has the property that the generalized matrix function is multiplicative on it and it is maximal with respect to the inclusion of transvections. It is further shown, with mild restrictions on the underlying field, that every matrix semigroup with the above properties is obtained in this way.

Isometries of symmetric gauge functions

Denote by F the field R of real numbers, the field C of complex numbers or the skew-field H of real quaternions. Let F n be the linear space of all column vectors with n coordinates over F A norm on F n is a symmetric gauge function if for any , for all permutations (i 1,…,i n) of (l,…,n), and for all possible choices of . The purpose of this note is to characterize the isometries of symmetric gauge functions on F n . In particular, we show that G is the isometry group of a symmetric gauge function on F n if and only if G is a compact subgroup of GL n (F) containing P n (F), the group of generalized permutation matrices over F. To prove the main theorem, we establish some results concerning the groups that contain P n (F), and characterize all finite subgroups of GL n (R) that contain P n (R), which are of independent interest. While most of these results could be deduced from the advanced theories of Lie groups and reflection groups, elementary proofs are given to them in this paper.

On linear MDS codes of length q + 1 over GF( q) for even q

Monomial matrices are constructed which transform doubly-extended Reed-Solomon codes into BCH codes of length q + 1 over GF( q), where q is a power of 2.

Hadamard matrices of order 36 with automorphisms of order 17

A Hadamard matrix of order n is an n by n matrix of 1’s and − 1’s such that HHt − nI. In such a matrix n is necessarily 1, 2 or a multiple of 4. Two Hadamard matrices H1 and H2 are called equivalent if there exist monomial matrices P, Q with PH1Q = H2. An automorphism of a Hadamard matrix H is an equivalence of the matrix to itself, i.e. a pair (P, Q) of monomial matrices such that PHQ = H. In other words, an automorphism of H is a permutation of its rows followed by multiplication of some rows by − 1, which leads to reordering of its columns and multiplication of some columns by − 1. The set of all automorphisms form a group under composition called the automorphism group (Aut H) of H. For a detailed study of the basic properties and applications of Hadamard matrices see, e.g. [1], [7, Chap. 14], [8].

Normalizer of the subgroup of diagonal matrices in the general linear group over a ring

The normalizer of the diagonal subgroup in the general linear group over certain classes of rings is determined. Sufficient conditions are given under which this normalizer coincides with the group of monomial matrices.

Examples of nonmonomial linear groups without transvections

Z. I. Borevich (Ref. Zh. Mat., 1977, 5A284) gave a description of the subgroup H of GL(n, K) containing all of the diagonal matrices, under the condition that the field K contains at least seven elements. It follows from this description that if H contains no transvections, then H is contained in the subgroup of monomial matrices. In this note examples are constructed to show that the last assertion is false if card K⩽5. This shows that the condition card K⩾7 in the above-mentioned paper is essential.

On p-rational monomial matrix groups and Hadamard matrices

On p-rational monomial matrix groups and Hadamard matrices

On p-rational monomial matrix groups and Hadamard matrices

On p-rational monomial matrix groups and Hadamard matrices

Linear transformations on symmetric matrices that preserve the permanent

It is shown that if a linear transformation T on the space of n-square symmetric matrices over any subfield of the real field preserves the permanent, where n ⩾ 3, then T( A)= ± PAP t for all symmetric matrices A and a fixed generalized permutation matrix P with per P= ± 1.