The sign matrix and the separation of matrix eigenvalues

The sign matrices uniquely associated with the matrices (M ? ζ_jI)², where ζ_j are the corners of a rectangle oriented at π/4 to the axes of a Cartesian coordinate system, may be used to compute the number of eigenvalues of the arbitrarily chosen matrix M which lie within the rectangle, and to determine the left and right invariant subspaces of M associated with these eigenvalues. This paper is concerned with the proof of this statement, and with the details of the computation of the required sign matrices.     

The linear preservers of real diagonalizable matrices

Using a recent result of Bogdanov and Guterman on the linear preservers of pairs of simultaneously diagonalizable matrices, we determine all the automorphisms of the vector space M_n(R) which stabilize the set of diagonalizable matrices. To do so, we investigate the structure of linear subspaces of diagonalizable matrices of M_n(R) with maximal dimension.     

Ordering of nonnegative definite matrices with application to comparison of linear models

For any real nonnegative definite matrices M₁ and M₂ several necessary and sufficient conditions for M₁ ? M₂ are presented. The conditions are used for the comparison of linear models.     

Generalizations of Pauli channels

The Pauli channel acting on 2 × 2 matrices is generalized to an n-level quantum system. When the full matrix algebra M _n is decomposed into pairwise complementary subalgebras, then trace-preserving linear mappings M _n → M _n are constructed such that the restriction to the subalgebras are depolarizing channels. The result is the necessary and sufficient condition of complete positivity. The main examples appear on bipartite systems.     

Linear maps preserving M–P inverses of matrices between matrix spaces over fields

Suppose F is a field of characteristic not 2. Let n and m be two arbitrary positive integers with n≥2. We denote by M _n (F) and S _n (F) the space of n×n full matrices and the space of n×n symmetric matrices over F, respectively. All linear maps from S _n (F) to M _m (F) preserving M–P inverses of matrices are characterized first, and thereby all linear maps from S _n (F) (M _n (F)) to S _m (F) (M _m (F)) preserving M–P inverses of matrices are characterized, respectively.     

A problem on conjugacy of matrices

Let F be an infinite field and n?12. Then the number of conjugacy classes of the upper triangular nilpotent matrices in M_n(F) under action by the subgroup of GL_n(F) consisting of all the upper triangular matrices is infinite.     

Homomorphisms of matrix semigroups over division rings from dimension two to four

Let D be an arbitrary division ring and M_n(D) the multiplicative semigroup of all n×n matrices over D. We study non-degenerate, injective homomorphisms from M₂(D) to M₄(D). In particular, we present a structural result for the case when D is the ring of quaternions.     

Subspaces of matrices with special rank properties

Let K be a field and let M_m×n(K) denote the space of m×n matrices over K. We investigate properties of a subspace M of M_m×n(K) of dimension n(m-r+1) in which each non-zero element of M has rank at least r and enumerate the number of elements of a given rank in M when K is finite. We also provide an upper bound for the dimension of a constant rank r subspace of M_m×n(K) when K is finite and give non-trivial examples to show that our bound is optimal in some cases. We include a similar a bound for the maximum dimension of a constant rank subspace of skew-symmetric matrices over a finite field.     

Author index to volume 20

This paper considers the conjecture that given a real nonsingular matrix A, there exist a real diagonal matrix Λ with $\text{¦λ}{}_{\mathrm{ii}}\text{λ = 1}$ and a permutation matrix P such that (ΛPA) is positive stable. The conjecture is shown to be true for matrices of order 3 or less and may not be true for higher order matrices. A counterexample is presented in terms of a matrix of order 65. In showing this, an interesting matrix M_l of order 2^l = 64, which satisfies the matrix equation 2^l-1(M_l + M^T_l), has been used. The stability analysis is done by first decomposing the nonsingular matrix into its polar form. Some interesting results are presented in the study of eigenvalues of a product of orthogonal matrices. A simple function is derived in terms of these orthogonal matrices, which traces a hysteresis loop.     

Eigenvalue assignments and the two largest multiplicities in a Hermitian matrix whose graph is a tree

Among the possible multiplicity lists for the eigenvalues of Hermitian matrices whose graph is a tree we focus upon M₂, the maximum value of the sum of the two largest multiplicities. The corresponding M₁ is already understood. The notion of assignment (of eigenvalues to subtrees) is formalized and applied. Using these ideas, simple upper and lower bounds are given for M₂ (in terms of simple graph theoretic parameters), cases of equality are indicated, and a combinatorial algorithm is given to compute M₂ precisely. In the process, several techniques are developed that likely have more general uses.     

Linear preservers of row-dense matrices

Let M_m,n be the set of all m × n real matrices. A matrix A ∈ M_m,n is said to be row-dense if there are no zeros between two nonzero entries for every row of this matrix. We find the structure of linear functions T: M_m,n → M_m,n that preserve or strongly preserve row-dense matrices, i.e., T(A) is row-dense whenever A is row-dense or T(A) is row-dense if and only if A is row-dense, respectively. Similarly, a matrix A ∈ M_n,m is called a column-dense matrix if every column of A is a column-dense vector. At the end, the structure of linear preservers (strong linear preservers) of column-dense matrices is found.     

Inner derivations of simple Lie pencils of rank 1

We prove that simple Lie pencils of rank 1 over an algebraically closed field P of characteristic 0 with operators of left multiplication being derivations are of the form of a sandwich algebra M ₃(U,D′), where U is the subspace of all skew-symmetric matrices in M ₃(P) and D′ is any subspace containing 〈E〉 in the space of all diagonal matrices D in M ₃(P).     

Singularity conditions for the non-existence of a common quadratic Lyapunov function for pairs of third order linear time invariant dynamic systems

The condition that a finite collection of stable matrices {A₁, … , A_M} has no common quadratic Lyapunov function (CQLF) is formulated as a hierarchy of singularity conditions for block matrices involving a number of unknown parameters. These conditions are applied to the case of two stable 3 × 3 matrices, where they are used to derive necessary and sufficient conditions for the non-existence of a CQLF.     

Congruence of Hermitian matrices by Hermitian matrices

Two Hermitian matrices A,B∈M_n(C) are said to be Hermitian-congruent if there exists a nonsingular Hermitian matrix C∈M_n(C) such that B=CAC. In this paper, we give necessary and sufficient conditions for two nonsingular simultaneously unitarily diagonalizable Hermitian matrices A and B to be Hermitian-congruent. Moreover, when A and B are Hermitian-congruent, we describe the possible inertias of the Hermitian matrices C that carry the congruence. We also give necessary and sufficient conditions for any 2-by-2 nonsingular Hermitian matrices to be Hermitian-congruent. In both of the studied cases, we show that if A and B are real and Hermitian-congruent, then they are congruent by a real symmetric matrix. Finally we note that if A and B are 2-by-2 nonsingular real symmetric matrices having the same sign pattern, then there is always a real symmetric matrix C satisfying B=CAC. Moreover, if both matrices are positive, then C can be picked with arbitrary inertia.     

Matrix analysis for associated consistency in cooperative game theory

Hamiache axiomatized the Shapley value as the unique solution verifying the inessential game property, continuity and associated consistency. Driessen extended Hamiache’s axiomatization to the enlarged class of efficient, symmetric, and linear values. In this paper, we introduce the notion of row (resp. column)-coalitional matrix in the framework of cooperative game theory. The Shapley value as well as the associated game are represented algebraically by their coalitional matrices called the Shapley standard matrix M^Sh and the associated transformation matrix M_λ, respectively. We develop a matrix approach for Hamiache’s axiomatization of the Shapley value. The associated consistency for the Shapley value is formulated as the matrix equality M^Sh = M^Sh · M_λ. The diagonalization procedure of M_λ and the inessential property for coalitional matrices are fundamental tools to prove the convergence of the sequence of repeated associated games as well as its limit game to be inessential. In addition, a similar matrix approach is applicable to study Driessen’s axiomatization of a certain class of linear values. In summary, it is illustrated that matrix analysis is a new and powerful technique for research in the field of cooperative game theory.     

Boolean designs and self-dual matroids

A proper splitting of a rectangular matrix A is one of the form A = M ? N, where A and M have the same range and null spaces. This concept was introduced by R. Plemmons as a means of generalizing to rectangular and singular matrices the concept of a regular splitting of a nonsingular matrix as introduced by R. Varga. In consideration of the linear system Ax=b, A. Berman and R. Plemmons used a proper splitting of A into M ? N and showed that the iteration x⁽ⁱ⁺¹⁾=M⁺Nx⁽ⁱ⁾+M⁺b converges to A⁺b, the best least-squares solution to the system, if and only if the spectral radius of M⁺N is less than one. The purpose of this paper is to further develop the characteristics of proper splittings and to extend these previous results by replacing the Moore-Penrose generalized inverse with a least-squares g-inverse, a minimum-norm g-inverse, or a g-inverse. Also, some criteria are given for comparing convergence rates of M_i^?N_i, where A = M₁?N₁ = M₂?N₂, and a method is developed for constructing proper splittings of special types of matrices.     

A theorem on products of matrices

A new result on products of matrices is proved in the following theorem: let M_i (i=1,2,…) be a bounded sequence of square matrices, and K be the l.u.b. of the spectral radii ρ(M_i). Then for any positive number ε there is a constant A and an ordering p(j) (j = 1,2,…) of the matrices such that

$\text{∏}\text{j=1}\text{n}\text{M}{}_{\mathrm{p(j)}}\text{?A·(K+ε)}{}^{\mathrm{n}}\text{(n = 1,2,…)}$

. The ordering is well defined by p(j), a one-to-one mapping on the set of positive integers. In general the inequality does not hold for any ordering p(j) (a counterexample is provided); however, some sufficient conditions are given for the result to remain true irrespective of the order of the matrices.     

Identities of symmetric and skew-symmetric matrices in characteristicp

It is well known how the Kostant-Rowen Theorem extends the validity of the famous Amitsur-Levitzki identity to skew-symmetric matrices. Here we give a general method, based on a graph theoretic approach, for deriving extensions of known permanental-type identities to skew-symmetric and symmetric matrices over a commutative ring of prime characteristic. Our main result has a typical Kostant-Rowen flavour: IfM≥p[n+1/2] then $C_M (X,Y) = \sum\limits_{\alpha ,\beta \in Sym(M)} {x_{\alpha (1)} y_{\beta (1)} x_{\alpha (2)} y_{\beta (2)} } ...x_{\alpha (M)} y_{\beta (M)} = 0$ is an identity onM _n ^? (Ω), the set ofnxn skew-symmetric matrices over a commutative ring Ω withp1_Ω=0 (provided that $P > \sqrt {[n + 1/2)} $ ). Otherwise, the stronger conditionM≥pn implies thatC _M(X,Y)=0 is an identity on the full matrix ringM _n(Ω).