Topic
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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Dissertation•
01 Oct 2007
TL;DR: In this article, the authors consider the quadratic eigenvalue problem with singular leading and trailing coefficients and present two types of algorithms: linearization and Householder reflectors.
Abstract: In this thesis we consider algorithms for solving the quadratic eigenvalue problem,
(lambda^2*A_2 + lambda*A_1 + A_0)x=0
when the leading or trailing
coefficient matrices are singular. In a finite element discretization this corresponds to the mass or stiffness matrices
being singular
and reflects modes of vibration (or eigenvalues) at zero or ``infinity''. We are interested in deflation procedures
that enable us to utilize knowledge of the presence of these (or any) eigenvalues to reduce the overall cost in
computing the remaining eigenvalues and eigenvectors of interest.
We first give an introduction to the quadratic eigenvalue problem and explain how it can be solved by a process called linearization.
We present two types of algorithms, firstly a modification of an algorithm published by
Kublanovskaya, Mikhailov, and Khazanov in the 1970s that has recently been translated into English.
Using these ideas we present algorithms that are able to reduce the size of the problem by ``deflating''
infinite and zero eigenvalues that arise when the mass or stiffness matrix (or both) are singular.
Secondly we look at methods that deflate zero and infinite eigenvalues by the use of Householder reflectors;
this requires a basis for the null space of the mass or stiffness matrix (or both), so we also summarize various decompositions
that can be used to give this information.
We consider different applications that yield a quadratic eigenvalue problem
with singular leading and trailing coefficients and after testing the implementations of the algorithms
on some of these problems we comment on their stability.
TL;DR: In this article, a new short proof using properties of the field of values was given to show that a complex matrix with only real eigenvalues is either hermitian or has indefinite imaginary part.
Abstract: We give a new short proof using properties of the field of values to show that a) a complex matrix with only real eigenvalues is either hermitian or has indefinite imaginary part, and b) one with only purely imaginary eigenvalues is either skew-hermitian or has indefinite real part, while c) one whose eigenvalues all have absolute value 1 is either unitary or has indefinite polar defect I—TT* . Conversely, every skewsymmetric matrix is the skewsymmetric part of some real matrix that is similar to a real diagonal matrix. The corresponding result for complex matrices is found to be false.
TL;DR: In this paper, the authors apply a theorem of Ky Fan on eigenvalue location to give a new proof of a result of K. J. Palmer concerning the real parts of the eigenvalues of a matrix.
Abstract: We apply a theorem of Ky Fan on eigenvalue location to give a new proof of a result of K. J. Palmer concerning the real parts of the eigenvalues of a matrix.
TL;DR: In this paper, a Koehler-type method to obtain lower bounds for the eigenvalues of a certain class of operators is presented, and general properties required of a problem for the technique to work are discussed and the connection with other classical methods is analyzed.
Abstract: A Koehler-type method to obtain lower bounds for the eigenvalues of a certain class of operators is presented. The general properties required of a problem for the technique to work are discussed and the connection with other classical methods is analyzed.