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There exist efficient programs for bringing a diagonalizable matrix to diagonal form by a similarity transformation. In theory of congruence transformations, unitoid matrices are analogs of diagonalizable matrices. However, excepting Hermitian and, more generally, normal matrices, there are no recognized programs for bringing a unitoid matrix to diagonal form by a congruence transformation. We propose an algorithm that is able to perform this task for a special class of unitoid matrices, namely, nonsingular matrices whose cosquares are well-conditioned with respect to the complete eigenproblem. Examples are presented to illustrate the performance of the algorithm. [ABSTRACT FROM AUTHOR]

Let A = (aij) and H = (hij) be positive semidefinite matrices of the same order. If aij ≥ |hij| for all i, j; A is diagonally dominant and all row sums of H are equal to zero, then we show that the sum of all k x k principal minors of A is greater than or equal to the sum of all k x k principal minors of H. [ABSTRACT FROM AUTHOR]

In this paper, according to the theory of two classes of $ \alpha $-diagonally dominant matrices, the row index set of the matrix is divided properly, and then some positive diagonal matrices are constructed. Furthermore, some new criteria for nonsingular $ H $-matrix are obtained. Finally, numerical examples are given to illustrate the effectiveness of the proposed criteria.

In this paper, according to the theory of two classes of -diagonally dominant matrices, the row index set of the matrix is divided properly, and then some positive diagonal matrices are constructed. Furthermore, some new criteria for nonsingular -matrix are obtained. Finally, numerical examples are given to illustrate the effectiveness of the proposed criteria. [ABSTRACT FROM AUTHOR]

We evaluate the dual cone of the set of diagonally dominant matrices (resp., scaled diagonally dominant matrices), namely DD n ∗ (resp., SDD n ∗ ), as an approximation of the semidefinite cone. We prove that the norm normalized distance, proposed by Blekherman et al. [5], between a set S and the semidefinite cone has the same value whenever SDD n ∗ ⊆ S ⊆ DD n ∗ . This implies that the norm normalized distance is not a sufficient measure to evaluate these approximations. As a new measure to compensate for the weakness of that distance, we propose a new distance, called the trace normalized distance. We prove that the trace normalized distance between DD n ∗ and S + n has a different value from the one between SDD n ∗ and S + n and give the exact values of these distances. [ABSTRACT FROM AUTHOR]

In this paper, we study the class of relatively D -stable matrices and provide sufficient conditions for relative D -stability. We generalize the well-known Hadamard inequality, to provide upper bounds for the determinants of relatively D -stable and relatively additive D -stable matrices. For some classes of D -stable matrices, we estimate the sector gap between matrix spectra and the imaginary axis. We apply the developed technique to obtain upper bounds for determinants of some classes of D -stable matrices, e.g. diagonally stable, diagonally dominant and matrices with Q 2 -scalings. [ABSTRACT FROM AUTHOR]

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In this paper, we study the dissipative property of the first order $ 2\times 2 $ hyperbolic system with constant coefficients. We propose a dissipative condition (see (2.9)) which is weaker than the strongly dissipative condition and can be regarded as a generalization of Kawashima-Shizuta condition. We show that this condition is sharp. With this condition and tools of Fourier analysis, we also give pointwise estimates of the solution to the Cauchy problem for suitable initial data. Finally, we illustrate that our dissipative condition can not be generalized directly to $ 3\times3 $ system. [ABSTRACT FROM AUTHOR]