New classes of spectrally arbitrary ray patterns

An n×n ray pattern matrix S is said to be spectrally arbitrary if for every monic nth degree polynomial f(λ) with coefficients from C, there is a complex matrix in the ray pattern class of S such that its characteristic polynomial is f(λ). In this article we give new classes of spectrally arbitrary ray pattern matrices.     

Several spectrally arbitrary ray patterns

A ray pattern A of order n is said to be spectrally arbitrary if given any monic nth degree polynomial f(x) with coefficients from ?, there exists a matrix realization of A such that its characteristic polynomial is f(x). An n?×?n ray pattern A is said to be minimally spectrally arbitrary if replacing any nonzero entry of A by zero destroys this property. In this article, several families of ray patterns are presented and proved to be minimally spectrally arbitrary. We also show that for n?≥?5, when A _n is spectrally arbitrary, then it is minimally spectrally arbitrary.     

Hoffman polynomials of nonnegative irreducible matrices and strongly connected digraphs

For a nonnegative n × n matrix A, we find that there is a polynomial f(x)∈R[x] such that f(A) is a positive matrix of rank one if and only if A is irreducible. Furthermore, we show that the lowest degree such polynomial f(x) with tr f(A) = n is unique. Thus, generalizing the well-known definition of the Hoffman polynomial of a strongly connected regular digraph, for any irreducible nonnegative n × n matrix A, we are led to define its Hoffman polynomial to be the polynomial f(x) of minimum degree satisfying that f(A) is positive and has rank 1 and trace n. The Hoffman polynomial of a strongly connected digraph is defined to be the Hoffman polynomial of its adjacency matrix. We collect in this paper some basic results and open problems related to the concept of Hoffman polynomials.     

Potentially nilpotent and spectrally arbitrary even cycle sign patterns

An n × n sign pattern S_n is potentially nilpotent if there is a real matrix having sign pattern S_n and characteristic polynomial xⁿ. A new family of sign patterns C_n with a cycle of every even length is introduced and shown to be potentially nilpotent by explicitly determining the entries of a nilpotent matrix with sign pattern C_n. These nilpotent matrices are used together with a Jacobian argument to show that C_n is spectrally arbitrary, i.e., there is a real matrix having sign pattern C_n and characteristic polynomial for any real μ_i. Some results and a conjecture on minimality of these spectrally arbitrary sign patterns are given.     

Unary operations with long pre-periods

It is well known that the congruence lattice ConA of an algebra A is uniquely determined by the unary polynomial operations of A (see e.g. [K. Denecke, S.L. Wismath, Universal Algebra and Applications in Theoretical Computer Science, Chapman & Hall, CRC Press, Boca Raton, London, New York, Washington DC, 2002 [2]]). Let A be a finite algebra with |A|=n. If Imf=A or |Imf|=1 for every unary polynomial operation f of A, then A is called a permutation algebra. Permutation algebras play an important role in tame congruence theory [D. Hobby, R. McKenzie, The structure of finite algebras, Contemporary Mathematics, vol. 76, Providence, Rhode Island, 1988 [3]]. If f:A→A is not a permutation then A⊃Imf and there is a least natural number λ(f) with Imf^λ(f)=Imf^λ(f)+1. We consider unary operations with λ(f)=n-1 for n?2 and λ(f)=n-2 for n?3 and look for equivalence relations on A which are invariant with respect to such unary operations. As application we show that every finite group which has a unary polynomial operation with one of these properties is simple or has only normal subgroups of index 2.     

On the span of Hadamard products of vectors

Let A₁, … , A_k be positive semidefinite matrices and B₁, … , B_k arbitrary complex matrices of order n. We show that

span{(A₁x)°(A₂x)°?°(A_kx)|x∈Cⁿ}=range(A₁°A₂°?°A_k)     

A new family of spectrally arbitrary ray patterns

An n × n ray pattern A is called a spectrally arbitrary ray pattern if the complex matrices in Q(A) give rise to all possible complex polynomials of degree n.     

Topological classification of affine operators on unitary and Euclidean spaces

We study affine operators on a unitary or Euclidean space U up to topological conjugacy. An affine operator is a map f:U→U of the form f(x)=Ax+b, in which A:U→U is a linear operator and b∈U. Two affine operators f and g are said to be topologically conjugate if g=h^-1fh for some homeomorphism h:U→U.If an affine operator f(x)=Ax+b has a fixed point, then f is topologically conjugate to its linear part A. The problem of classifying linear operators up to topological conjugacy was studied by Kuiper and Robbin [Topological classification of linear endomorphisms, Invent. Math. 19 (2) (1973) 83-106] and other authors.Let f:U→U be an affine operator without fixed point. We prove that f is topologically conjugate to an affine operator g:U→U such that U is an orthogonal direct sum of g-invariant subspaces V and W,

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the restriction g∣V of g to V is an affine operator that in some orthonormal basis of V has the form

(x₁,x₂,…,x_n)?(x₁+1,x₂,…,x_n-1,εx_n)     

Inequalities between ∥f(A + B)∥ and ∥f(A) + f(B)∥

The conjecture posed by Aujla and Silva [J.S. Aujla, F.C. Silva, Weak majorization inequalities and convex functions, Linear Algebra Appl. 369 (2003) 217-233] is proved. It is shown that for any m-tuple of positive-semidefinite n × n complex matrices A_j and for any non-negative convex function f on [0, ∞) with f(0) = 0 the inequality ?f(A₁) + f(A₂) + ? + f(A_m)? ? ? f(A₁ + A₂ + ? + A_m)? holds for any unitarily invariant norm ? · ?. It is also proved that ?f(A₁) + f(A₂) + ? + f(A_m)? ? f(?A₁ + A₂ + ? + A_m?), where f is a non-negative concave function on [0, ∞) and ? · ? is normalized.     

To what extent is a large space of matrices not closed under product?

Let (K) be a field. Given an arbitrary linear subspace V of M_n(K) of codimension less than n-1, a classical result states that V generates the (K)-algebra M_n(K). Here, we strengthen this statement in three ways: we show that M_n(K) is spanned by the products of the form AB with (A,B)∈V²; we prove that every matrix in M_n(K) can be decomposed into a product of matrices of V; finally, when V is a linear perplane of M_n(K) and n>2, we show that every matrix in M_n(K) is a product of two elements of V.     

幂零--中心化方法

An nxn complex sign pattern(ray pattern) S is said to be spectrally arbitrary if for every monic nth degree polynomial f(λ) with coefficients from C,there is a complex matrix in the complex sign pattern class(ray pattern class) of S such that its characteristic polynomial is f(λ).We derive the Nilpotent-Centralizer methods for spectrally arbitrary complex sign patterns and ray patterns,respectively.We find that the Nilpotent-Centralizer methods for three kinds of patterns(sign pattern,complex sign pattern,ray pattern) are the same in form.     

A generalized Cayley-Hamilton theorem for polynomials with matrix coefficients

A family F of square matrices of the same order is called a quasi-commuting family if (AB-BA)C=C(AB-BA) for all A,B,C∈F where A,B,C need not be distinct. Let f_k(x₁,x₂,…,x_p),(k=1,2,…,r), be polynomials in the indeterminates x₁,x₂,…,x_p with coefficients in the complex field C, and let M₁,M₂,…,M_r be n×n matrices over C which are not necessarily distinct. Let and let δ_F(x₁,x₂,…,x_p)=detF(x₁,x₂,…,x_p). In this paper, we prove that, for n×n matrices A₁,A₂,…,A_p over C, if {A₁,A₂,…,A_p,M₁,M₂,…,M_r} is a quasi-commuting family, then F(A₁,A₂,…,A_p)=O implies that δ_F(A₁,A₂,…,A_p)=O.     

Max algebraic powers of irreducible matrices in the periodic regime: An application of cyclic classes

In max algebra it is well known that the sequence of max algebraic powers A^k, with A an irreducible square matrix, becomes periodic after a finite transient time T(A), and the ultimate period γ is equal to the cyclicity of the critical graph of A.In this connection, we study computational complexity of the following problems: (1) for a given k, compute a periodic power A^r with and r?T(A), (2) for a given x, find the ultimate period of {A^l⊗x}. We show that both problems can be solved by matrix squaring in O(n³logn) operations. The main idea is to apply an appropriate diagonal similarity scaling A?X^-1AX, called visualization scaling, and to study the role of cyclic classes of the critical graph.     

Absolutely indecomposable symmetric matrices

Let A be a symmetric matrix of size n×n with entries in some (commutative) field K. We study the possibility of decomposing A into two blocks by conjugation by an orthogonal matrix T∈Mat_n(K). We say that A is absolutely indecomposable if it is indecomposable over every extension of the base field. If K is formally real then every symmetric matrix A diagonalizes orthogonally over the real closure of K. Assume that K is a not formally real and of level s. We prove that in Mat_n(K) there exist symmetric, absolutely indecomposable matrices iff n is congruent to 0, 1 or −1 modulo 2s.     

Procrustes problems and associated approximation problems for matrices with k-involutory symmetries

We say that a matrix R∈C^n×n is k-involutary if its minimal polynomial is x^k-1 for some k?2, so R^k-1=R^-1 and the eigenvalues of R are 1,ζ,ζ²,…,ζ^k-1, where ζ=e^2πi/k. Let α,μ∈{0,1,…,k-1}. If R∈C^m×m, A∈C^m×n, S∈C^n×n and R and S are k-involutory, we say that A is (R,S,μ)-symmetric if RAS^-1=ζ^μA, and A is (R,S,α,μ)-symmetric if RAS^-α=ζ^μA.Let L be the class of m×n(R,S,μ)-symmetric matrices or the class of m×n(R,S,α,μ)-symmetric matrices. Given X∈C^n×t and B∈C^m×t, we characterize the matrices A in L that minimize ‖AX-B‖ (Frobenius norm), and, given an arbitrary W∈C^m×n, we find the unique matrix A∈L that minimizes both ‖AX-B‖ and ‖A-W‖. We also obtain necessary and sufficient conditions for existence of A∈L such that AX=B, and, assuming that the conditions are satisfied, characterize the set of all such A.     

A note on the perturbation of positive matrices by normal and unitary matrices

In a recent paper, Neumann and Sze considered for an n × n nonnegative matrix A, the minimization and maximization of ρ(A + S), the spectral radius of (A + S), as S ranges over all the doubly stochastic matrices. They showed that both extremal values are always attained at an n × n permutation matrix. As a permutation matrix is a particular case of a normal matrix whose spectral radius is 1, we consider here, for positive matrices A such that (A + N) is a nonnegative matrix, for all normal matrices N whose spectral radius is 1, the minimization and maximization problems of ρ(A + N) as N ranges over all such matrices. We show that the extremal values always occur at an n × n real unitary matrix. We compare our results with a less recent work of Han, Neumann, and Tastsomeros in which the maximum value of ρ(A + X) over all n × n real matrices X of Frobenius norm was sought.     

Preservers of eigenvalue inclusion sets

For a square matrix A, let S(A) be an eigenvalue inclusion set such as the Gershgorin region, the Brauer region in terms of Cassini ovals, and the Ostrowski region. Characterization is obtained for maps Φ on n×n matrices satisfying S(Φ(A)-Φ(B))=S(A-B) for all matrices A and B. From these results, one can deduce the structure of additive or (real) linear maps satisfying S(A)=S(Φ(A)) for every matrix A.     

Invertible and nilpotent matrices over antirings

In this paper, we characterize invertible matrices over an arbitrary commutative antiring S with 1 and find the structure of GL_n(S). We find the number of nilpotent matrices over an entire commutative finite antiring. We prove that every nilpotent n×n matrix over an entire antiring can be written as a sum of ⌈log₂n⌉ square-zero matrices and also find the necessary number of square-zero summands for an arbitrary trace-zero matrix to be expressible as their sum.