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

Let λ j and a j j , 1 ≤ j ≤ n , be the eigenvalues and the diagonal entries of a Hermitian matrix A , both enumerated in the increasing order. We prove some inequalities that are stronger than the Schur majorisation inequalities ∑ j = 1 r λ j ≤ ∑ j = 1 r a j j , 1 ≤ r ≤ n. [ABSTRACT FROM AUTHOR]

In this paper, we determine the underlying graphs of the mixed graphs with smallest Hermitian eigenvalue greater than − 3. Furthermore, we characterize all mixed graphs on n ≥ 11 vertices with smallest Hermitian eigenvalue greater than − 3. The mixed graphs on n ≤ 10 vertices with smallest Hermitian eigenvalue greater than − 3 could be also obtained easily with the help of computer because their underlying graphs are determined. Roughly speaking, we completely determine the mixed graphs with smallest Hermitian eigenvalue greater than − 3. [ABSTRACT FROM AUTHOR]

In the block version of the classical two-sided Jacobi method for the Hermitian eigenvalue problem, the off-diagonal elements of iterated matrix A (k) converge to zero. However, this fact alone does not necessarily guarantee that A (k) converges to a fixed diagonal matrix. The same is true for the matrix of accumulated unitary transformations Q (k). We prove that under certain assumptions A (k) indeed converges to a fixed diagonal matrix, whose diagonal elements are the eigenvalues of the input matrix A. Next it is shown that for a simple eigenvalue the corresponding column of Q (k) converges to the corresponding eigenvector. For a multiple eigenvalue or a cluster of eigenvalues, we prove that the orthogonal projectors constructed from the corresponding columns of Q (k) converge to the orthogonal projector onto the eigenspace corresponding to those eigenvalues. Moreover, the appropriate convergence bounds are obtained for all discussed cases. Convergence results are also valid for the parallel block-Jacobi method with dynamic ordering. The developed theory is illustrated by numerical example. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to prove the following determinantal inequality: det ⁡ (A k + | B s 2 A s 2 | 2 t s ) ≤ det ⁡ (A k + A t B t) ≤ det ⁡ (A k + | A s 2 B s 2 | 2 t s ) for any positive semi-definite matrices A and B , and for all 0 ≤ t ≤ s ≤ k. It generalizes several known determinantal inequalities, and one main consequence of it confirms Lin's conjecture which states that for positive semi-definite matrices A and B , det ⁡ (A 2 + A t B t) ≤ det ⁡ (A 2 + | A B | t) for 0 ≤ t ≤ 2. We conclude with another related determinantal inequality. [ABSTRACT FROM AUTHOR]

In 2017, M. Lin formulated two conjectures concerning determinantal inequalities for positive semi-definite matrices A and B , and which can be stated as follows det ⁡ (A 2 + | A B | p) ≥ det ⁡ (A 2 + | B A | p) for p ≥ 0 and det ⁡ (A 2 + | A B | p) ≥ det ⁡ (A 2 + A p B p) for 0 ≤ p ≤ 2. The main goal of this paper is to confirm the first conjecture in a slightly more general setting namely in the case when A and B are Hermitian, and also to prove the second conjecture when 0 ≤ p ≤ 4 3. Various related inequalities are then presented and we conclude with an open log-majorization question. [ABSTRACT FROM AUTHOR]

The singular values of a matrix A are defined as the square roots of the eigenvalues of A ⁎ A , and the energy of A denoted by E (A) is the sum of its singular values. The energy of a graph G , E (G) , is defined as the sum of absolute values of the eigenvalues of its adjacency matrix. In this paper, we prove that if A is a Hermitian matrix with the block form A = ( B D D ⁎ C ) , then E (A) ≥ 2 E (D). Also, we show that if G is a graph and H is a spanning subgraph of G such that E (H) is an edge cut of G , then E (H) ≤ E (G) , i.e., adding any number of edges to each part of a bipartite graph does not decrease its energy. Let G be a connected graph of order n and size m with the adjacency matrix A. It is well-known that if G is a bipartite graph, then E (G) ≥ 4 m + n (n − 2) | det ⁡ (A) | 2 n . Here, we improve this result by showing that the inequality holds for all connected graphs of order at least 7. Furthermore, we improve a lower bound for E (G) given in Oboudi (2019) [14]. [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]

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 show some majorization inequalities and apply them to derive norm, eigenvalue, singular value, and trace inequalities of matrices. We also present a generalization and a different proof of a recent result on majorization concerning partitioned Hermitian positive semidefinite matrices. [Copyright &y& Elsevier]

Abstract: Estimating upper bounds of the spectrum of large Hermitian matrices has long been a problem with both theoretical and practical significance. Algorithms that can compute tight upper bounds with minimum computational cost will have applications in a variety of areas. We present a practical algorithm that exploits k-step Lanczos iteration with a safeguard step. The k is generally very small, say 5–8, regardless of the large dimension of the matrices. This makes the Lanczos iteration economical. The safeguard step can be realized with marginal cost by utilizing the theoretical bounds developed in this paper. The bounds establish the theoretical validity of a previous bound estimator that has been successfully used in various applications. Moreover, we improve the bound estimator which can now provide tighter upper bounds with negligible additional cost. [Copyright &y& Elsevier]

Abstract: The inertia of a Hermitian matrix is defined to be a triplet composed of the numbers of the positive, negative and zero eigenvalues of the matrix counted with multiplicities, respectively. In this paper, we show some basic formulas for inertias of block Hermitian matrices. From these formulas, we derive various equalities and inequalities for inertias of sums, parallel sums, products of Hermitian matrices, submatrices in block Hermitian matrices, differences of outer inverses of Hermitian matrices. As applications, we derive the extremal inertias of the linear matrix expression with respect to a variable Hermitian matrix X. In addition, we give some results on the extremal inertias of Hermitian solutions to the matrix equation , as well as the extremal inertias of a partial block Hermitian matrix. [Copyright &y& Elsevier]

Abstract: In this paper, by using normal maps originated by Lewis [A.S. Lewis, Group invariance and convex matrix analysis, SIAM J. Matrix Anal. Appl. 17 (1996) 927–949], we present a method for producing majorization inequalities on eigenvalues and singular values of matrices. Some classical and new inequalities are derived. [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]

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]