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Spectrum of a matrix
About: Spectrum of a matrix is a research topic. Over the lifetime, 1064 publications have been published within this topic receiving 19841 citations. The topic is also known as: matrix spectrum.
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TL;DR: In this article, it was shown that the number of distinct eigenvalues of a matrix after perturbation can at most double the number after a rank-one update of the matrix.
Abstract: We prove a new theorem relating the number of distinct eigenvalues of a matrix after perturbation to the prior number of distinct eigenvalues, the rank of the update, and the degree of nondiagonalizability of the matrix. In particular, a rank one update applied to a diagonalizable matrix can at most double the number of distinct eigenvalues. The theorem applies to both symmetric and nonsymmetric matrices and perturbations, of arbitrary magnitudes. An an application, we prove that in exact arithmetic the number of Krylov iterations required to exactly solve a linear system involving a diagonalizable matrix can at most double after a rank one update.
1 citations
TL;DR: In this paper, a method was derived for the determination of the eigenvalues and corresponding eigenfunctions which arise in the problem of forced convection of heat through an infinite tube of arbitrary cross-section.
1 citations
TL;DR: It is shown that given this special structure and a certain sign condition, the dimension of the eigenvalue problem is greatly reduced, and some theorems on eigen Value crossing directions are given.
1 citations
16 Sep 2014
TL;DR: In this paper, a tridiagonal matrix with specified multiple eigenvalues was constructed from the viewpoint of the quotient difference recursion formula, and it was shown that the characteristic and minimal polynomials of a constructed tridiagon matrix are equal to each other.
Abstract: In this paper, we grasp an inverse eigenvalue problem which constructs a tridiagonal matrix with specified multiple eigenvalues, from the viewpoint of the quotient difference (qd) recursion formula. We also prove that the characteristic and the minimal polynomials of a constructed tridiagonal matrix are equal to each other. As an application of the qd formula, we present a procedure for getting a tridiagonal matrix with specified multiple eigenvalues. Examples are given through providing with four tridiagonal matrices with specified multiple eigenvalues.
1 citations
TL;DR: In this article, the majority of the eigenvalues of a graph are at most −1, thus disproving a conjecture of Yong (Linear Algebra Appl. 295 (1999) 73).
1 citations