Topic
Square matrix
About: Square matrix is a research topic. Over the lifetime, 5000 publications have been published within this topic receiving 92428 citations.
Papers published on a yearly basis
Papers
More filters
TL;DR: New algorithms that can replace the diagonal entries of a Hermitian matrix by any set of diagonal entries that majorize the original set without altering the eigenvalues of the matrix are presented.
Abstract: In this paper, we present new algorithms that can replace the diagonal entries of a Hermitian matrix by any set of diagonal entries that majorize the original set without altering the eigenvalues of the matrix. They perform this feat by applying a sequence of (N-1) or fewer plane rotations, where N is the dimension of the matrix. Both the Bendel--Mickey and the Chan--Li algorithms are special cases of the proposed procedures. Using the fact that a positive semidefinite matrix can always be factored as $\mtx{X^\adj X}$, we also provide more efficient versions of the algorithms that can directly construct factors with specified singular values and column norms. We conclude with some open problems related to the construction of Hermitian matrices with joint diagonal and spectral properties.
55 citations
TL;DR: A simple method which leads to results on the Lipschitz continuity of the matrix absolute value and several new ones is discussed.
55 citations
TL;DR: A method is presented based on combinatorial considerations which permutes the rows and columns of a general matrix in such a way that relatively dense blocks of various sizes appear along the diagonal.
Abstract: Block iterative methods used for the solution of linear systems of algebraic equations can perform better when the diagonal blocks of the corresponding matrix are carefully chosen. A method is presented based on combinatorial considerations which permutes the rows and columns of a general matrix in such a way that relatively dense blocks of various sizes appear along the diagonal. The method is particularly useful when no natural partitioning of the matrix is available. Two parameters govern the method which is $O(n +
u )$ in time and space, where n is the order of the matrix and $
u $ is the number of nonzeros in the matrix. Numerical test results are presented which illustrate the performance of both the ordering algorithm and the block iterative methods with the resulting orderings.
55 citations
TL;DR: In this article, it was shown that the total variation distance of the random variable Tr(AM) to a standard normal random variable is bounded by 2√3 n-1 and this rate is sharp up to the constant.
Abstract: Let M be a random matrix in the orthogonal group On, distributed according to Haar measure, and let A be a fixed n x n matrix over R such that Tr(AA t ) = n. Then the total variation distance of the random variable Tr(AM) to a standard normal random variable is bounded by 2√3 n-1 and this rate is sharp up to the constant. Analogous results are obtained for M a random unitary matrix and A a fixed n x n matrix over C. The proofs are applications of a new abstract normal approximation theorem which extends Stein's method of exchangeable pairs to situations in which continuous symmetries are present.
55 citations
TL;DR: In this article, the authors derived an alternative representation for MD under the same assumptions, but with the condition on the Schur complement in the hypothesis replaced by the condition that R (CAA D ) ⊂ N (B ) ∩ N (D ), where R ( · ) and N ( · ), are the range and null space of a matrix.
55 citations