Monomial Matrices Research Articles

Linear Transformations On Matrices: The Invariance of Generalized Permutation Matrices, I

Let F be a field, Mn(F) be the vector space of all w-square matrices with entries in F and a subset of Mn(F). It is of interest to determine the structure of linear maps T : Mn(F) →Mn(F) such that . For example: Let be GL(n, C), the group of all nonsingular n X n matrices over C [5]; the subset of all rank 1 matrices in MmXn(F) [4] (MmXn(F) is the vector space of all m X n matrices over F) ; the unitary group [2] ; or the set of all matrices X in Mn(F) such that det(X) = 0 [1]. Other results in this direction can be found in [3].

Linear transformations on matrices: the invariance of generalized permutation matrices—III

Let F be a field, F∗ be its multiplicative group, and H = { H: H is a subgroup of F∗ and there do not exist a, bϵ F∗ such that Ha+ b⊆ H}. Let D n be the dihedral group of degree n, H be a nontrivial group in H , and τ n ( H) = {α = (α 1, α 2,…, α n ):α i ϵ H}. For σϵ D n and αϵτ n ( H), let P(σ, α) be the matrix whose ( i,j) entry is α i δ iσ( j ) (i.e., a generalized permutation matrix), and P(D n, H) = {P(σ, α):σϵD n, αϵτ n(H)} . Let M n ( F) be the vector space of all n× n matrices over F and T P( D n , H) = { T: T is a linear transformation on M n ( F) to itself and T( P( D n , H)) = P( D n , H)}. In this paper we classify all T in T P( D n , H) and determine the structure of the group T P( D n , H) (Theorems 1 to 4). An expository version of the main results is given in Sec. 1, and an example is given at the end of the paper.

Linear transformations on nonnegative matrices

It is shown that if a linear transformation T on the space of n × n complex matrices maps nonnegative matrices into nonnegative matrices and preserves the spectrum of each nonnegative matrix, then T( A) = P −1 AP or T( A) = P −1 A T P for all matrices A and a fixed nonnegative generalized permutation matrix P.

Generalized Semigroups of Quotients

For S a semigroup with 0 and ${M_S}$ a right S-set, certain classes of sub S-sets called right quotient filters are defined. A study of these right quotient filters is made and examples are given including the classes of intersection large and dense sub S-sets respectively. The general semigroup of right quotients Q corresponding to a right quotient filter on a semigroup S is developed and basic properties of this semigroup are noted. A nonzero regular semigroup S is called primitive dependent if each nonzero right ideal of S contains a 0-minimal right ideal of S. The theory developed in the paper enables us to characterize all primitive dependent semigroups having singular congruence the identity in terms of subdirect products of column monomial matrix semigroups over groups.

Generalized semigroups of quotients

For S a semigroup with 0 and M S {M_S} a right S-set, certain classes of sub S-sets called right quotient filters are defined. A study of these right quotient filters is made and examples are given including the classes of intersection large and dense sub S-sets respectively. The general semigroup of right quotients Q corresponding to a right quotient filter on a semigroup S is developed and basic properties of this semigroup are noted. A nonzero regular semigroup S is called primitive dependent if each nonzero right ideal of S contains a 0-minimal right ideal of S. The theory developed in the paper enables us to characterize all primitive dependent semigroups having singular congruence the identity in terms of subdirect products of column monomial matrix semigroups over groups.

Controllability and Observability in Linear Time-Variable Networks With Arbitrary Symmetry Groups

This paper presents a unified treatment of linear time-variable networks displaying arbitrary geometrical symmetries by incorporating group theory into an analysis scheme. Symmetric networks have their elements arranged so that certain permutations of the network edges result in a configuration which is identical with the original. These permutations lead to a group of monomial matrices which are shown to commute with the network A-matrix and the state transition matrix of the normal form equation. The representation theory of groups facilitates the study of those network properties which are determined solely by symmetry. By using group theory, a simple arithmetic condition is derived which, when satisfied, implies that the network is noncontrollable or nonobservable because of symmetry alone. The results allow the determination by inspection of linear combinations of the original state variables which result in noncontrollable variables. It is shown that networks displaying axial point group symmetry are generally only weakly controllable.

Symmetry codes over GF(3) and new five-designs

For every odd prime power q where q ≡ −1(3) we define a (2 q + 2, q + 1) code over the field of three elements. It is shown that all the codes in this family are self orthogonal. For q = 5, the (12, 6) code is equivalent to the extended Golay code. For q = 11, it can be shown that the minimum weight of the (24, 12) code is 9. For q = 17, 23, 29 it is shown, in part by computer, that the minimum weights of the (36, 18), (48, 24), and (60, 30) codes are 12, 15, and 18 respectively. There are 5-designs associated with vectors of certain weights in the (12, 6), (24, 12), (36, 18), (48, 24), and (60, 30) codes. There are new 5-designs associated with the last four codes mentioned. The 5-designs related to the (36, 18) and (60, 30) codes are the first 5-designs found with their parameters. For each q we construct a group P of (2 q + 2) × (2 q + 2) monomial matrices. We show that P leaves the (2 q + 2, q + 1) code in the family invariant, and that P {I, −I} is isomorphic to PGL 2( q). We can form a Hadamard matrix by considering the rows of this matrix as certain maximal weight vectors contained in this code. This Hadamard matrix is left invariant by the group P described above.

Hadamard matrices of order cube plus one

2. Definitions and proof. Throughout this paper 7 and / denote the identity matrix and the matrix with 1 in every position respectively, of the order required by the context. An a, b matrix is one in which each element is either a or b. An Hadamard matrix is a 1, —1 matrix 77 of order h such that HHT = hI. (Necessarily either h = 2 or h is divisible by 4.) It is of type 1 if 77+77r = 27. An Hadamard design A is a 0, 1 matrix of order h — such that AAT = ATA = (/j/4)7+(V4-1)J. (Necessarily^ J = JA = (h/2-\)J.) It is of type 1 if A+AT = J-I. If 77 is an Hadamard matrix it can be multiplied by generalized permutation matrices to bring it into the form