On the Courant-Fischer theory for Krein spaces

Let J=I_r⊕-I_n-r,0<r<n. An n×n complex matrix A is said to be J-Hermitian if JA=A^∗J. An extension of the classical theory of Courant and Fischer on the Rayleigh ratio of Hermitian matrices is stated for J-Hermitian matrices. Applications to the theory of small oscilations of a mechanical system are presented.     

An extension of Birkhoff’s theorem with an application to determinants

Let M_n(F) denote the algebra of n×n matrices over the field F of complex, or real, numbers. Given a self-adjoint involution J∈M_n(C), that is, J=J^*,J²=I, let us consider Cⁿ endowed with the indefinite inner product [,] induced by J and defined by [x,y]?〈Jx,y〉,x,y∈Cⁿ. Assuming that (r,n-r), 0?r?n, is the inertia of J, without loss of generality we may assume J=diag(j₁,?,j_n)=I_r⊕-I_n-r. For T=(|t_ik|²)∈M_n(R), the matrices of the form T=(|t_ik|²j_ij_k), with all line sums equal to 1, are called J-doubly stochastic matrices. In the particular case r∈{0,n}, these matrices reduce to doubly stochastic matrices, that is, non-negative real matrices with all line sums equal to 1. A generalization of Birkhoff’s theorem on doubly stochastic matrices is obtained for J-doubly stochastic matrices and an application to determinants is presented.     

Nonlinear Lie derivations on upper triangular matrices

Let M _n (𝔸) and T _n (𝔸) be the algebra of all n?×?n matrices and the algebra of all n?×?n upper triangular matrices over a commutative unital algebra 𝔸, respectively. In this note we prove that every nonlinear Lie derivation from T _n (𝔸) into M _n (𝔸) is of the form A?→?AT???TA?+?A _??+?ξ(A)I _n , where T?∈?M _n (𝔸), ??:?𝔸?→?𝔸 is an additive derivation, ξ?:?T _n (𝔸)?→?𝔸 is a nonlinear map with ξ(AB???BA)?=?0 for all A,?B?∈?T _n (𝔸) and A _? is the image of A under???applied entrywise.     

A characterisation of common diagonal stability over cones

In this article, a general notion of common diagonal Lyapunov matrix is formulated for a collection of n?×?n matrices A ₁,?…?,?A _s , and cones k ₁,?…?,?k _s in ? ⁿ . Necessary and sufficient conditions are derived for the existence of a common diagonal Lyapunov matrix in this setting. The conditions are similar to and extend the well-known criteria for the case s?=?1, k ₁?=?? ⁿ .     

Minimum permanents of doubly stochastic matrices with at least one zero entry †

It is shown that the minimum value of the permanent on the n× ndoubly stochastic matrices which contain at least one zero entry is achieved at those matrices nearest to J_n in Euclidean norm, where J_n is the n× nmatrix each of whose entries is n^-1 . In case n ≠ 3 the minimum permanent is achieved only at those matrices nearest J_n ; for n= 3 it is achieved at other matrices containing one or more zero entries as well.     

The inverse problem of bisymmetric matrices with a submatrix constraint

An n×n real matrix A is called a bisymmetric matrix if A=A^T and A=S_nAS_n, where S_n is an n×n reverse unit matrix. This paper is mainly concerned with solving the following two problems: Problem I Given n×m real matrices X and B, and an r×r real symmetric matrix A₀, find an n×n bisymmetric matrix A such that where A([1: r]) is a r×r leading principal submatrix of the matrix A. Problem II Given an n×n real matrix A^*, find an n×n matrix  in S_E such that where ∥·∥ is Frobenius norm, and S_E is the solution set of Problem I. The necessary and sufficient conditions for the existence of and the expressions for the general solutions of Problem I are given. The explicit solution, a numerical algorithm and a numerical example to Problem II are provided. Copyright © 2003 John Wiley & Sons, Ltd.     

Distributions of eigenvalues and inertias of some block Hermitian matrices consisting of orthogonal projectors

A complex square matrix A is called an orthogonal projector if A ²?=?A?=?A*, where A* is the conjugate transpose of A. In this article, we first give some formulas for calculating the distributions of real eigenvalues of a linear combination of two orthogonal projectors. Then, we establish various expansion formulas for calculating the inertias, ranks and signatures of some 2?×?2 and 3?×?3, as well as k?×?k block Hermitian matrices consisting of two orthogonal projectors. Many applications of the formulas are presented in characterizing interval distributions of numbers of eigenvalues, and nonsingularity of these block Hermitian matrices. In addition, necessary and sufficient conditions are given for various equalities and inequalities of these block Hermitian matrices to hold.     

Simultaneous similarity of matrices

In this paper we solve completely and explicitly the long-standing problem of classifying pairs of n × n complex matrices (A, B) under the simultaneous similarity (TAT⁻¹, TBT⁻¹). Roughly speaking, the classification decomposes to a finite number of steps. In each step we consider an open algebraic set ⁰_n,2,r,π M_n × M_n (M_n = the set of n × n complex-valued matrices). Here r and π are two positive integers. Then we construct a finite number of rational functions ø₁,…,ø_s in the entries of A and B whose values are constant on all pairs similar in _n,2,r,π to (A, B). The values of the functions ø_i(A, B), I = 1,…, s, determine a finite number (at most κ(n, 2, r)) of similarity classes in _n,2,r,π. Let S_n be the subspace of complex symmetric matrices in M_n. For (A, B) ε S_n × S_n we consider the similarity class (TAT^t, TBT^t), where T ranges over all complex orthogonal matrices. Then the characteristic polynomial |λI − (A + xB)| determines a finite number of similarity classes for almost all pairs (A, B) ε S_n × S_n.     

Uniform and minimal {0,1} – cp matrices

Let A be an n?×?n real matrix. A is called {0,1}-cp if it can be factorized as A?=?BB ^T with b_ij =0 or 1. The smallest possible number of columns of B in such a factorization is called the {0,1}-rank of A. A {0,1}-cp matrix A is called minimal if for every nonzero nonnegative n?×?n diagonal matrix D, A-D is not {0,1}-cp, and r-uniform if it can be factorized as A=BB ^T, where B is a (0,?1) matrix with r 1s in each column. In this article, we first present a necessary condition for a nonsingular matrix to be {0,1}-cp. Then we characterize r-uniform {0,1}-cp matrices. We also obtain some necessary conditions and sufficient conditions for a matrix to be minimal {0,1}-cp, and present some bounds for {0,1}-ranks.     

Idempotence preserving maps between matrix spaces

Suppose 𝔽 is an arbitrary field of characteristic not 2 and 𝔽?≠?𝔽₃. Let M _n (𝔽) be the space of all n?×?n full matrices over 𝔽 and P _n (𝔽) the subset of M _n (𝔽) consisting of all n?×?n idempotent matrices and GL _n (𝔽) the subset of M _n (𝔽) consisting of all n?×?n invertible matrices. Let Φ_𝔽(n,?m) denote the set of all maps from M _n (𝔽) to M _m (𝔽) satisfying A???λB?∈?P _n (𝔽)???φ(A)???λφ(B)?∈?P _m (𝔽) for every A,?B?∈?M _n (𝔽) and λ?∈?𝔽, where m and n are integers with 3?≤?n?≤?m. It is shown that if φ?∈?Φ_𝔽(n,?m), then there exists T?∈?GL _m (𝔽) such that φ(A)?=?T?[A???I _p ?⊕?A ^t ???I _q ?⊕?0]T?^?1 for every A?∈?M _n (𝔽), where I ₀?=?0. This improves the results of some related references.     

The solution of the equation AX + BX ⋆ = 0

We give a complete solution of the matrix equation AX?+?BX ^??=?0, where A, B?∈?? ^m×n are two given matrices, X?∈?? ^n×n is an unknown matrix, and ? denotes the transpose or the conjugate transpose. We provide a closed formula for the dimension of the solution space of the equation in terms of the Kronecker canonical form of the matrix pencil A?+?λB, and we also provide an expression for the solution X in terms of this canonical form, together with two invertible matrices leading A?+?λB to the canonical form by strict equivalence.     

Function of a square matrix with repeated eigenvalues

An analytical function f(A) of an arbitrary n×n constant matrix A is determined and expressed by the “fundamental formula”, the linear combination of constituent matrices. The constituent matrices Z_kh, which depend on A but not on the function f(s), are computed from the given matrix A, that may have repeated eigenvalues. The associated companion matrix C and Jordan matrix J are then expressed when all the eigenvalues with multiplicities are known. Several other related matrices, such as Vandermonde matrix V, modal matrix W, Krylov matrix K and their inverses, are also derived and depicted as in a 2-D or 3-D mapping diagram. The constituent matrices Z_kh of A are thus obtained by these matrices through similarity matrix transformations. Alternatively, efficient and direct approaches for Z_kh can be found by the linear combination of matrices, that may be further simplified by writing them in “super column matrix” forms. Finally, a typical example is provided to show the merit of several approaches for the constituent matrices of a given matrix A.     

On linear combinations of generalized involutive matrices

Let X ^? denotes the Moore--Penrose pseudoinverse of a matrix X. We study a number of situations when (aA?+?bB)^??=?aA?+?bB provided a,?b?∈?????{0} and A, B are n?×?n complex matrices such that A ^??=?A and B ^??=?B.     

Large spaces of symmetric matrices of bounded rank are decomposable ∗

Let k and n be positive integers such that kn. Let S_n (F) denote the space of all n×n symmetric matrices over the field F with char F≠2. A subspace L of S_n (F) is said to be a k-subspace if rank Ak for every A?L.

Now suppose that k is even, and write k=2r. We say a k∥-subspace of S_n (F) is decomposable if there exists in Fⁿ a subspace W of dimension n?r such that x^tAx=0 for every x?W A?L.

We show here, under some mild assumptions on k n and F, that every k∥-subspace of S_n (F) of sufficiently large dimension must be decomposable. This is an analogue of a result obtained by Atkinson and Lloyd for corresponding subspaces of F^m,n .     

Jacobson radical of skew polynomial rings and skew group rings

LetR be a ring and σ an automorphism ofR. We prove the following results: (i)J(R _σ[x])={Σ_ir_i x ⁱ:r₀∈I∩J(R]), r _i∈I for alliε 1} whereI↪ {r∈R:rx ∈J(R _Σ[x])|s= (ii)J(R _σ<x>)=(J(R _σ<x>)∩R)_σ<x>. As an application of the second result we prove that ifG is a solvable group such thatG andR, + have disjoint torsions thenJ(R)=0 impliesJ(R(G))=0.     

A bound on the scrambling index of a primitive matrix using Boolean rank

The scrambling index of an n×n primitive matrix A is the smallest positive integer k such that A^k(A^t)^k=J, where A^t denotes the transpose of A and J denotes the n×n all ones matrix. For an m×n Boolean matrix M, its Boolean rank b(M) is the smallest positive integer b such that M=AB for some m×b Boolean matrix A and b×n Boolean matrix B. In this paper, we give an upper bound on the scrambling index of an n×n primitive matrix M in terms of its Boolean rank b(M). Furthermore we characterize all primitive matrices that achieve the upper bound.     

Linear preservers and quantum information science

In this article, a brief survey of recent results on linear preserver problems and quantum information science is given. In addition, characterization is obtained for linear operators φ on mn?×?mn Hermitian matrices such that φ(A???B) and A???B have the same spectrum for any m?×?m Hermitian A and n?×?n Hermitian B. Such a map has the form A???B???U(?₁(A)????₂(B))U* for mn?×?mn Hermitian matrices in tensor form A???B, where U is a unitary matrix, and for j?∈?{1,?2}, ? _j is the identity map?X???X or the transposition map?X???X ^t . The structure of linear maps leaving invariant the spectral radius of matrices in tensor form A???B is also obtained. The results are connected to bipartite (quantum) systems and are extended to multipartite systems.