Volume 2, Issue 1

Letr Σ n (C) denote the space of all n χ n symmetric matrices over the complex field C. The main objective of this paper is to prove that the maps Φ : Σ n (C) -> Σ n (C) satisfying for any fixed irre- ducible characters X, X' -S C the condition d x (A +aB) = d χ ·(Φ(Α ) + αΦ(Β)) for all matrices A,В ε Σ„(С) and all scalars a ε C are automatically linear and bijective. As a corollary of the above result we characterize all such maps Φ acting on Σ И (С).

We prove that any quaternionic matrix of order n ≤3 admits a characteristic function, whose roots are the left eigenvalues, that satisfes Cayley-Hamilton theorem.

Define n × n tridiagonal matrices T and S as follows: All entries of the main diagonal of T are zero and those of the first super- and subdiagonal are one. The entries of the main diagonal of S are two except the (n, n) entry one, and those of the first super- and subdiagonal are minus one. Then, denoting by λ(·) the largest eigenvalue, Using certain lower bounds for the largest eigenvalue, we provide lower bounds for these expressions and, further, lower bounds for sin x and cos x on certain intervals. Also upper bounds can be obtained in this way.

The zero forcing number and the positive zero forcing number of a graph are two graph parameters that arise from two types of graph colourings. The zero forcing number is an upper bound on the minimum number of induced paths in the graph that cover all the vertices of the graph, while the positive zero forcing number is an upper bound on the minimum number of induced trees in the graph needed to cover all the vertices in the graph. We show that for a block-cycle graph the zero forcing number equals the path cover number.We also give a purely graph theoretical proof that the positive zero forcing number of any outerplanar graphs equals the tree cover number of the graph. These ideas are then extended to the setting of k-trees, where the relationship between the positive zero forcing number and the tree cover number becomes more complex.

This is a survey of recent results concerning (integer) matrices whose leading principal minors are well-known sequences such as Fibonacci, Lucas, Jacobsthal and Pell (sub)sequences. There are different ways for constructing such matrices. Some of these matrices are constructed by homogeneous or nonhomogeneous recurrence relations, and others are constructed by convolution of two sequences. In this article, we will illustrate the idea of these methods by constructing some integer matrices of this type.

Let A be an invertible 3 × 3 complex matrix. It is shown that there is a 3 × 3 permutation matrix P such that the product PA has at least two distinct eigenvalues. The nilpotent complex n × n matrices A for which the products PA with all symmetric matrices P have a single spectrum are determined. It is shown that for a n × n complex matrix there exists a permutation matrix P such that the product PA has at least two distinct eigenvalues.

We generalize the theory of positive diagonal scalings of real positive definite matrices to complex diagonal scalings of complex positive definite matrices. A matrix A is a diagonal scaling of a positive definite matrix M if there exists an invertible complex diagonal matrix D such that A = D*MD and where every row and every column of A sums to one. We look at some of the key properties of complex diagonal scalings and we conjecture that every n by n positive definite matrix has at most 2 n−1 scalings and prove this conjecture for certain special classes of matrices.We also use the theory of complex diagonal matrix scalings to formulate a van der Waerden type question on the permanent function; we show that the solution of this question would have applications to finding certain maximally entangled quantum states.

An n × n diamond alternating sign matrix (ASM) is a (0, +1, −1)-matrix with ±1 entries alternatingand arranged in a diamond-shaped pattern. The explicit inverse (for n even) or generalized inverse (for n odd) of a diamond ASM is derived. The eigenvalues of diamond ASMs are considered and when n is even, thecharacteristic polynomial, which involves signed binomial coefficients, is determined.

We supply a combinatorial description of any minor of the adjacency matrix of a graph. This descriptionis then used to give a formula for the determinant and inverse of the adjacency matrix, A(G) , of agraph G , whenever A(G) is invertible, where G is formed by replacing the edges of a tree by path bundles.

Companion matrices of the second type are characterized by properties that involve bilinear maps.

Let P ∈ ℂ mxm and Q ∈ ℂ n×n be invertible matrices partitioned as P = [P 0 P 1 · · · P k−1 ] and Q = [Q 0 Q 1 · · · Q k−1 ], with P ℓ ∈ ℂ m×mℓ and Q ℓ ∈ ℂ n×nℓ , 0 ≤ ℓ ≤ k − 1. Partition P −1 and Q −1 as where P̂ ℓ ∈ ℂ mℓ ×m , Q̂ ℓ ∈ ℂ nℓ×n , P̂ℓP m = δ ℓm I mℓ , and Q̂ ℓ Q m = δ ℓm I nℓ , 0 ≤ ℓ, m ≤ k − 1. Let Z k = {0, 1, . . . , k − 1}. We study matrices A = P σ(ℓ) F ℓ Q ℓ and B = Q ℓ G ℓ P σ(ℓ) , where σ : Z k → Z k . Special cases: A = and B = , where A ℓ ∈ ℂ d1×d2 and B ℓ ∈ ℂ d2×d1 , 0 ≤ ℓ ≤ k − 1.

In this work we revisited some properties of Sylvester Hadamard matrices of order 2 k . Based onlyon the existence of a base from which any Sylvester Hadamard matrix can be constructed, we prove that theirrows (columns) are closed under addition and that the numbers of sign interchanges along any row (column)give all integers 0, 1, . . . , 2 k − 1 in some order. These two properties can be used to check if an Hadamardmatrix is equivalent to a Sylvester Hadamard matrix.

In this paper, a new characterization of previously studied generalized complementary basic matrices is obtained. It is in terms of ranks and structure ranks of submatrices defined by certain diagonal positions. The results concern both the irreducible and general cases

A family T (ν) , ν ∈ ℝ, of semiinfinite positive Jacobi matrices is introduced with matrix entries takenfrom the Hahn-Exton q-difference equation. The corresponding matrix operators defined on the linear hullof the canonical basis in ℓ 2 (ℤ + ) are essentially self-adjoint for |ν| ≥ 1 and have deficiency indices (1, 1) for|ν| < 1. A convenient description of all self-adjoint extensions is obtained and the spectral problem is analyzedin detail. The spectrum is discrete and the characteristic equation on eigenvalues is derived explicitlyin all cases. Particularly, the Hahn-Exton q-Bessel function J ν (z; q) serves as the characteristic function ofthe Friedrichs extension. As a direct application one can reproduce, in an alternative way, some basic resultsabout the q-Bessel function due to Koelink and Swarttouw.

We give extensions of inequalities of Araki-Lieb-Thirring, Audenaert, and Simon, in the context ofsemisimple Lie groups.

Alon and Yuster give for independent identically distributed real or vector valued random variablesX, Y combinatorially proved estimates of the form Prob(∥X − Y∥ ≤ b) ≤ c Prob(∥X − Y∥ ≤ a). We derivethese using copositive matrices instead. By the same method we also give estimates for the real valued case,involving X + Y and X − Y, due to Siegmund-Schultze and von Weizsäcker as generalized by Dong, Li andLi. Furthermore, we formulate a version of the above inequalities as an integral inequality for monotonefunctions.

A matrix A ∈ ℝ n×n is a GM-matrix if A = sI − B, where 0 < ρ(B) ≤ s and B ∈WPFn i.e., both B and B t have ρ(B) as their eigenvalues and their corresponding eigenvector is entry wise nonnegative. In this article,we consider a generalization of a subclass of GM-matrices having a nonnegative core nilpotent decompositionand prove a characterization result for such matrices. Also, we study various notions of splitting of matricesfrom this new class and obtain sufficient conditions for their convergence.

L. Huang [Consimilarity of quaternion matrices and complex matrices, Linear Algebra Appl. 331(2001) 21–30] gave a canonical form of a quaternion matrix with respect to consimilarity transformationsA ↦ ˜S −1 AS in which S is a nonsingular quaternion matrix and h = a + bi + cj + dk ↦ ˜h := a − bi + cj − dk (a, b, c, d ∈ ℝ).We give an analogous canonical form of a quaternion matrix with respect to consimilarity transformationsA ↦^S −1 AS in which h ↦ ^h is an arbitrary involutive automorphism of the skew field of quaternions. Weapply the obtained canonical form to the quaternion matrix equations AX −^X B = C and X − A^X B = C.

Determinant formulas for special binary circulant matrices are derived and a new open problemregarding the possible determinant values of these specific circulant matrices is stated. The ideas used forthe proofs can be utilized to obtain more determinant formulas for other binary circulant matrices, too. Thesuperiority of the proposed approach over the standard method for calculating the determinant of a generalcirculant matrix is demonstrated.

Identified are certain special periodic diagonal matrices that have a predictable number of pairedeigenvalues. Since certain symmetric Toeplitz matrices are special cases, those that have several multiple 5eigenvalues are also investigated further. This work generalizes earlier work on response matrices from circularlysymmetric models.