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We classify maximal digraphs whose Hermitian spectral radius is at most 2. [ABSTRACT FROM AUTHOR]

Yoopyo Hong proved in 1989 that each Hermitian matrix A is orthogonally *congruent to a matrix of the form ε 1 A 1 ⊕ ⋯ ε r A r ⊕ B 1 ⊕ ⋯ ⊕ B s , in which A 1 , ... , A r , B 1 , ... , B s are uniquely determined by the orthogonal *congruence class of A, and ε 1 , ... , ε r ∈ { 1 , − 1 }. We prove that ε 1 , ... , ε p are uniquely determined by the orthogonal *congruence class of A as well. As an application, we present a canonical form of a pair (A , B) consisting of a Hermitian matrix A and a nonsingular symmetric matrix B with respect to transformations (S ⁎ A S , S T B S) with a nonsingular S. [ABSTRACT FROM AUTHOR]

Abstract We investigate the change in the multiplicities of the eigenvalues of a Hermitian matrix with a simple graph G , when edges are inserted into G or removed from G. We focus upon cases in which the multiplicity of the eigenvalue does not change due to inserting or removing edges incident to a vertex. Furthermore, we show how the change in the multiplicities of the eigenvalues occur, when two disjoint graphs are connected with one edge, based upon the status of the vertices that are connected. Lastly, we give the possible classifications of cut-edges in a graph and characterize the occurrence of each possible status. [ABSTRACT FROM AUTHOR]

For a Hermitian matrix A ∈ C p × p , given vectors x , y ∈ C p and for suitable functions f , the bilinear form y ⁎ f ( A ) x is estimated by extending the extrapolation method proposed by C. Brezinski in 1999. Families of one term and two term estimates e f , ν , ν ∈ C and e ˆ f , n , k , n , k ∈ Z , respectively, are derived by extrapolation of the moments of the matrix A . For the positive definite case, bounds for the optimal value of ν , which leads to an efficient one term estimate in only one matrix vector product, are derived. For f ( A ) = A − 1 , a formula approximating this optimal value of ν is specified. Numerical results for several matrix functions and comparisons are provided to demonstrate the effectiveness of the extrapolation method. [ABSTRACT FROM AUTHOR]

The enhanced principal rank characteristic sequence (epr-sequence) of a symmetric n × n matrix is a sequence ℓ 1 ℓ 2 ⋯ ℓ n where ℓ k is A , S , or N according as all, some, or none of its principal minors of order k are nonzero. Such sequences give more information than the (0,1) pr-sequences previously studied (where basically the k th entry is 0 or 1 according as none or at least one of its principal minors of order k is nonzero). Various techniques including the Schur complement are introduced to establish that certain subsequences such as NAN are forbidden in epr-sequences over fields of characteristic not two. Using probabilistic methods over fields of characteristic zero, it is shown that any sequence of A s and S s ending in A is attainable, and any sequence of A s and S s followed by one or more N s is attainable; additional families of attainable epr-sequences are constructed explicitly by other methods. For real symmetric matrices of orders 2, 3, 4, and 5, all attainable epr-sequences are listed with justifications. [ABSTRACT FROM AUTHOR]

We consider the problem of determining those undirected n -vertex graphs with a corresponding Hermitian matrix that admits only two distinct eigenvalues, with multiplicities k and n--k. After giving some general algebraic characterizations of these dual multiplicity graphs, we then prove two major graph theoretic necessary conditions on such graphs. Construction techniques are then developed, and these lead to a characterization of dual multiplicity graphs for which the lesser multiplicity is two. [ABSTRACT FROM AUTHOR]

Abstract: Let be a positive semidefinite block matrix with square matrices M and N of the same order and denote . The main results are the following eigenvalue majorization inequalities: for any complex number z of modulus 1, If, in addition, K is Hermitian, then for any real number , while if K is skew-Hermitian, then for any real number , where O is the zero matrix of compatible size. These majorization inequalities generalize some results due to Furuichi and Lin, Turkmen, Paksoy and Zhang, Lin and Wolkowicz. [Copyright &y& Elsevier]

Abstract: We propose a new QR-like algorithm, symmetric squared QR (SSQR) method, that can be readily parallelized using commonly available parallel computational primitives such as matrix–matrix multiplication and QR decomposition. The algorithm converges quadratically and the quadratic convergence is achieved through a squaring technique without utilizing any kind of shifts. We provide a rigorous convergence analysis of SSQR and derive structures for several of the important quantities generated by the algorithm. We also discuss various practical implementation issues such as stopping criteria and deflation techniques. We demonstrate the convergence behavior of SSQR using several numerical examples. [Copyright &y& Elsevier]

Baksalary and Pukelsheim [Linear Algebra Appl. 151 (1991) 135] considered the problem of how an order between two Hermitian nonnegative definite matrices A and B is related to the corresponding order between the squares A2 and B2, in the sense of the star partial ordering, the minus partial ordering, and the Lo¨wner partial ordering. In the present paper, possibilities of generalizing and strengthening their results are studied from two points of view: by widening the class of matrices considered and by replacing the squares by arbitrary powers. [Copyright &y& Elsevier]

We describe the set of all (α,n), for which the scalar complex matrix αIn is a sum of k idempotent Hermitian matrices, and get the minimal number of summands for each αIn. [Copyright &y& Elsevier]

For Hermitian matrices, whose graph is a given tree, the relationships among vertex degrees, multiple eigenvalues and the relative position of the underlying eigenvalue in the ordered spectrum are discussed in detail. In the process, certain aspects of special vertices, whose removal results in an increase in multiplicity are investigated. [Copyright &y& Elsevier]