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This paper presents two main results that the singular values of the Hadamard product of normal matrices Ai are weakly log-majorized by the singular values of the Hadamard product of |Ai| and the singular values of the sum of normal matrices Ai are weakly log-majorized by the singular values of the sum of |Ai|. Some applications to these inequalities are also given. In addition, several related and new inequalities are obtained. [ABSTRACT FROM AUTHOR]

Given three n-tuples {λi}ni =1, {μi}ni =1, {υi}ni =1 of complex numbers, we introduce the problem of when there exists a pair of normal matrices A and B such that σ(A) = {λi}ni =1, σ(B) = {μi}ni =1, and σ(A + B) = {υi}ni =1, where σ(.) denote the spectrum. In the case when λk = 0, k = 2, . . . , n, we provide necessary and sufficient conditions for the existence of A and B. In addition, we show that the solution pair (A,B) is unique up to unitary similarity. The necessary and sufficient conditions reduce to the classical A. Horn inequalities when the n-tuples are real. [ABSTRACT FROM AUTHOR]

In this note we characterize polynomial numerical hulls of matrices A ∈ Mn such that A² is Hermitian. Also, we consider normal matrices A ∈ Mn whose kth power are semidefinite. For such matrices we show that Vk (A) = σ(A). [ABSTRACT FROM AUTHOR]

Two theorems are proved on the condensed forms with respect to unitary similarity and congruence transformations. They provide a theoretical basis for constructing economical iterative methods for systems of linear equations whose matrices are low-rank perturbations of normal and conjugate-normal matrices. [ABSTRACT FROM AUTHOR]

Let s 1, ..., s n be arbitrary complex scalars. It is required to construct an n × n normal matrix A such that s i is an eigenvalue of the leading principal submatrix A i , i = 1, 2, ..., n. It is shown that, along with the obvious diagonal solution diag( s 1, ..., s n ), this problem always admits a much more interesting nondiagonal solution A. As a rule, this solution is a dense matrix; with the diagonal solution, it shares the property that each submatrix A i is itself a normal matrix, which implies interesting connections between the spectra of the neighboring submatrices A i and A i + 1. [ABSTRACT FROM AUTHOR]

The method MINRES-CN was earlier proposed by the authors for solving systems of linear equations with conjugate-normal coefficient matrices. It is now shown that this method is also applicable even if the coefficient matrix, albeit not conjugate-normal, is a low-rank perturbation of a symmetric matrix. If the perturbed matrix is still conjugate-normal, then, starting from some iteration step, the recursion underlying MINRES-CN becomes a three-term relation. These results are proved in terms of matrix condensed forms with respect to unitary congruences. [ABSTRACT FROM AUTHOR]

Two minimal residual methods for solving linear systems of the form (α U + β I) x = b, where U is a unitary matrix, are compared numerically. The first method uses conventional Krylov subspaces, while the second involves generalized Krylov subspaces. Experiments favor the second method if |α| > |β|. Moreover, the greater the ratio |α|/|β|, the higher the superiority of the second method. [ABSTRACT FROM AUTHOR]