On the Hermitian positive definite solutions of nonlinear matrix equation X + A∗XA = Q

Nonlinear matrix equation X^s + A∗X^−tA = Q, where A, Q are n × n complex matrices with Q Hermitian positive definite, has widely applied background. In this paper, we consider the Hermitian positive definite solutions of this matrix equation with two cases: s ? 1, 0 < t ? 1 and 0 < s ? 1, t ? 1. We derive necessary conditions and sufficient conditions for the existence of Hermitian positive definite solutions for the matrix equation and obtain some properties of the solutions. We also propose iterative methods for obtaining the extremal Hermitian positive definite solution of the matrix equation. Finally, we give some numerical examples to show the efficiency of the proposed iterative methods.     

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.     

Alexandrov’s inequality and conjectures on some Toeplitz matrices

We study determinant inequalities for certain Toeplitz-like matrices over C. For fixed n and N ? 1, let Q be the n × (n + N − 1) zero-one Toeplitz matrix with Q_ij = 1 for 0 ? j − i ? N − 1 and Q_ij = 0 otherwise. We prove that det(QQ^∗) is the minimum of det(RR^∗) over all complex matrices R with the same dimensions as Q satisfying ∣R_ij∣ ? 1 whenever Q_ij = 1 and R_ij = 0 otherwise. Although R has a Toeplitz-like band structure, it is not required to be actually Toeplitz. Our proof involves Alexandrov’s inequality for polarized determinants and its generalizations. This problem is motivated by Littlewood’s conjecture on the minimum 1-norm of N-term exponential sums on the unit circle. We also discuss polarized Bazin-Reiss-Picquet identities, some connections with k-tree enumeration, and analogous conjectured inequalities for the elementary symmetric functions of QQ^∗.     

Strong linear preservers of rank reverse permutability on triangular matrices

Let F be a field with ∣F∣ > 2 and T_n(F) be the set of all n × n upper triangular matrices, where n ? 2. Let k ? 2 be a given integer. A k-tuple of matrices A₁, …, A_k ∈ T_n(F) is called rank reverse permutable if rank(A₁ A₂ ? A_k) = rank(A_k A_k−1 ? A₁). We characterize the linear maps on T_n(F) that strongly preserve the set of rank reverse permutable matrix k-tuples.     

Positive definite solutions of the nonlinear matrix equation X + AXA = Q (q > 0)

In this paper, the nonlinear matrix equation X + A^∗X^qA = Q (q > 0) is investigated. Some necessary and sufficient conditions for existence of Hermitian positive definite solutions of the nonlinear matrix equations are derived. An effective iterative method to obtain the positive definite solution is presented. Some numerical results are given to illustrate the effectiveness of the iterative methods.     

Matrix inverse problem and its optimal approximation problem for R-skew symmetric matrices

Let R ∈ C^n×n be a nontrivial involution, i.e., R² = I and R ≠ ±I. A matrix A ∈ C^n×n is called R-skew symmetric if RAR = −A. The least-squares solutions of the matrix inverse problem for R-skew symmetric matrices with R∗ = R are firstly derived, then the solvability conditions and the solutions of the matrix inverse problem for R-skew symmetric matrices with R∗ = R are given. The solutions of the corresponding optimal approximation problem with R∗ = R for R-skew symmetric matrices are also derived. At last an algorithm for the optimal approximation problem is given. It can be seen that we extend our previous results [G.X. Huang, F. Yin, Matrix inverse problem and its optimal approximation problem for R-symmetric matrices, Appl. Math. Comput. 189 (2007) 482-489] and the results proposed by Zhou et al. [F.Z. Zhou, L. Zhang, X.Y. Hu, Least-square solutions for inverse problem of centrosymmetric matrices, Comput. Math. Appl. 45 (2003) 1581-1589].     

On single and double Soules matrices

Until now the concept of a Soules basis matrix of sign patternN consisted of an orthogonal matrix R∈R^n,n, generated in a certain way from a positive n-vector, which has the property that for any diagonal matrix Λ = diag(λ₁, … , λ_n), with λ₁ ? ? ? λ_n ? 0, the symmetric matrix A = RΛR^T has nonnegative entries only. In the present paper we introduce the notion of a pair of double Soules basis matrices of sign patternN which is a pair of matrices (P, Q), each in R^n,n, which are not necessarily orthogonal and which are generated in a certain way from two positive vectors, but such that PQ^T = I and such that for any of the aforementioned diagonal matrices Λ, the matrix A = PΛQ^T (also) has nonnegative entries only. We investigate the interesting properties which such matrices A have.As a preamble to the above investigation we show that the iterates, , generated in the course of the QR-algorithm when it is applied to A = RΛR^T, where R is a Soules basis matrix of sign pattern N, are again symmetric matrices generated by the Soules basis matrices R_k of sign pattern N which are themselves modified as the algorithm progresses.Our work here extends earlier works by Soules and Elsner et al.     

Minimal ranks of some quaternion matrix expressions with applications

Suppose that p(X, Y) = A − BX − X^(∗)B^(∗) − CYC^(∗) and q(X, Y) = A − BX + X^(∗)B^(∗) − CYC^(∗) are quaternion matrix expressions, where A is persymmetric or perskew-symmetric. We in this paper derive the minimal rank formula of p(X, Y) with respect to pair of matrices X and Y = Y^(∗), and the minimal rank formula of q(X, Y) with respect to pair of matrices X and Y = −Y^(∗). As applications, we establish some necessary and sufficient conditions for the existence of the general (persymmetric or perskew-symmetric) solutions to some well-known linear quaternion matrix equations. The expressions are also given for the corresponding general solutions of the matrix equations when the solvability conditions are satisfied. At the same time, some useful consequences are also developed.     

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.     

Operator decomposition of graphs and the reconstruction conjecture

We present the method of proving the reconstructibility of graph classes based on the new type of decomposition of graphs — the operator decomposition. The properties of this decomposition are described. Using this decomposition we prove the following. Let P and Q be two hereditary graph classes such that P is closed with respect to the operation of join and Q is closed with respect to the operation of disjoint union. Let M be a module of graph G with associated partition (A,B,M), where A∼M and B⁄∼M, such that G[A]∈P, G[B]∈Q and G[M] is not (P,Q)-split. Then the graph G is reconstructible.     

Bernstein operators for extended Chebyshev systems

Let U_n ⊂ Cⁿ[a, b] be an extended Chebyshev space of dimension n + 1. Suppose that f₀ ∈ U_n is strictly positive and f₁ ∈ U_n has the property that f₁/f₀ is strictly increasing. We search for conditions ensuring the existence of points t₀, …, t_n ∈ [a, b] and positive coefficients α₀, …, α_n such that for all f ∈ C[a, b], the operator B_n:C[a, b] → U_n defined by satisfies B_nf₀ = f₀ and B_nf₁ = f₁. Here it is assumed that p_n,k, k = 0, …, n, is a Bernstein basis, defined by the property that each p_n,k has a zero of order k at a and a zero of order n − k at b.     

Two-sided hyperbolic SVD

In this paper, we propose the two-sided hyperbolic SVD (2HSVD) for square matrices, i.e., A=UΣV^[∗], where U and V^[∗] are J-unitary (J=diag(±1)) and Σ is a real diagonal matrix of “double-hyperbolic” singular values. We show that, with some natural conditions, such decomposition exists without the use of hyperexchange matrices. In other words, U and V^[∗] are really J-unitary with regard to J and not some matrix which is permutationally similar to matrix J. We provide full characterization of 2HSVD and completely relate it to the semidefinite J-polar decomposition.     

k-Potence preserving maps without the linearity and surjectivity assumptions

Let M_n be the space of all n × n complex matrices, and let Γ_n be the subset of M_n consisting of all n × n k-potent matrices. We denote by Ψ_n the set of all maps on M_n satisfying A − λB ∈ Γ_n if and only if ?(A) − λ?(B) ∈ Γ_n for every A,B ∈ M_n and λ ∈ C. It was shown that ? ∈ Ψ_n if and only if there exist an invertible matrix P ∈ M_n and c ∈ C with c^k−1 = 1 such that either ?(A) = cPAP⁻¹ for every A ∈ M_n, or ?(A) = cPA^TP⁻¹ for every A ∈ M_n.     

On approximately simultaneously diagonalizable matrices

A collection A₁, A₂, …, A_k of n × n matrices over the complex numbers C has the ASD property if the matrices can be perturbed by an arbitrarily small amount so that they become simultaneously diagonalizable. Such a collection must perforce be commuting. We show by a direct matrix proof that the ASD property holds for three commuting matrices when one of them is 2-regular (dimension of eigenspaces is at most 2). Corollaries include results of Gerstenhaber and Neubauer-Sethuraman on bounds for the dimension of the algebra generated by A₁, A₂, …, A_k. Even when the ASD property fails, our techniques can produce a good bound on the dimension of this subalgebra. For example, we establish for commuting matrices A₁, …, A_k when one of them is 2-regular. This bound is sharp. One offshoot of our work is the introduction of a new canonical form, the H-form, for matrices over an algebraically closed field. The H-form of a matrix is a sparse “Jordan like” upper triangular matrix which allows us to assume that any commuting matrices are also upper triangular. (The Jordan form itself does not accommodate this.)     

Matrices A such that AA − AA are nonsingular

In this paper we study the class of square matrices A such that AA^† − A^†A is nonsingular, where A^† stands for the Moore-Penrose inverse of A. Among several characterizations we prove that for a matrix A of order n, the difference AA^† − A^†A is nonsingular if and only if R(A)⊕R(A^∗)=C_n,1, where R(·) denotes the range space. Also we study matrices A such that R(A)^⊥=R(A^∗).     

New oscillation criteria for linear matrix Hamiltonian systems

Some new oscillation criteria are established for the matrix linear Hamiltonian system X′=A(t)X+B(t)Y, Y′=C(t)X−A∗(t)Y under the hypothesis: A(t), B(t)=B∗(t)>0, and C(t)=C∗(t) are n×n real continuous matrix functions on the interval [t₀,∞), (−∞<t₀). These results are sharper than some previous results even for self-adjoint second order matrix differential systems.     

On extremal matrices of second largest exponent by Boolean rank

Let b = b(A) be the Boolean rank of an n × n primitive Boolean matrix A and exp(A) be the exponent of A. Then exp(A) ? (b − 1)² + 2, and the matrices for which equality occurs have been determined in [D.A. Gregory, S.J. Kirkland, N.J. Pullman, A bound on the exponent of a primitive matrix using Boolean rank, Linear Algebra Appl. 217 (1995) 101-116]. In this paper, we show that for each 3 ? b ? n − 1, there are n × n primitive Boolean matrices A with b(A) = b such that exp(A) = (b − 1)² + 1, and we explicitly describe all such matrices.     

On p-quasi-hyponormal operators

A Hilbert space operator A ∈ B(H) is said to be p-quasi-hyponormal for some 0 < p ? 1, A ∈ p − QH, if A^∗(∣A∣^2p − ∣A^∗∣^2p)A ? 0. If H is infinite dimensional, then operators A ∈ p − QH are not supercyclic. Restricting ourselves to those A ∈ p − QH for which A⁻¹(0) ⊆ A^∗-1(0), A ∈ p_∗ − QH, a necessary and sufficient condition for the adjoint of a pure p_∗ − QH operator to be supercyclic is proved. Operators in p_∗ − QH satisfy Bishop’s property (β). Each A ∈ p_∗ − QH has the finite ascent property and the quasi-nilpotent part H₀(A − λI) of A equals (A − λI)^-1(0) for all complex numbers λ; hence f(A) satisfies Weyl’s theorem, and f(A^∗) satisfies a-Weyl’s theorem, for all non-constant functions f which are analytic on a neighborhood of σ(A). It is proved that a Putnam-Fuglede type commutativity theorem holds for operators in p_∗ − QH.