Principal Submatrices, Geometric Multiplicities, and Structured Eigenvectors

Johnson, Charles R.; Kroschel, Brenda K.

doi:10.1137/s0895479894266568

Cited by 7 publications

(6 citation statements)

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“…Then, Lemma 1, applied to (2), shows that, for any t 6 ¼ 0 (t ¼ 0 being the only exception in Lemma 1), m AþtE ii ðÞ ¼ m AðiÞ ðÞ.…”

Section: Proofmentioning

confidence: 99%

“…For (b), Lemma 1, applied to (2), shows that m AþtE ii ðÞ ¼ m AðiÞ ðÞ, with the lone exception of t ¼ 0. Proof Suppose that m A(i) () ¼ m A ().…”

Section: Proofmentioning

confidence: 99%

“…Using the same expansion for the characteristic polynomial of A þ tE ii , t 2 F, we obtain p AþtE ii ðxÞ ¼ ðx À a ii À tÞ p AðiÞ ðxÞ þ qðxÞ ¼ p A ðxÞ À tp AðiÞ ðxÞ: ð2Þ (2) implies that is also a root of…”

mentioning

confidence: 98%

“…Our purpose here is several-fold: (1) to focus on a single multiple eigenvalue, but in the very general setting of square matrices over a field; (2) to understand the relationship between change in multiplicity with change in a diagonal entry and change in multiplicity when passing to a principal submatrix and (3) to apply these observations to get further, or different, insights in the Hermitian/tree case. Our basic results generalize certain aspects of the Hermitian case.…”

mentioning

confidence: 99%

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The change in eigenvalue multiplicity associated with perturbation of a diagonal entry

Johnson

Duarte

Saiago

2012

Linear and Multilinear Algebra

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“…Then, Lemma 1, applied to (2), shows that, for any t 6 ¼ 0 (t ¼ 0 being the only exception in Lemma 1), m AþtE ii ðÞ ¼ m AðiÞ ðÞ.…”

Section: Proofmentioning

confidence: 99%

“…For (b), Lemma 1, applied to (2), shows that m AþtE ii ðÞ ¼ m AðiÞ ðÞ, with the lone exception of t ¼ 0. Proof Suppose that m A(i) () ¼ m A ().…”

Section: Proofmentioning

confidence: 99%

mentioning

confidence: 98%

mentioning

confidence: 99%

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The change in eigenvalue multiplicity associated with perturbation of a diagonal entry

Johnson

Duarte

Saiago

2012

Linear and Multilinear Algebra

Self Cite

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“…NONCONVEX QUADRATIC OPTIMIZATION 405 We n o w use the fact that the set of permutation matrices is equal to the intersection of the orthogonal matrices with the 0,1 matrices. We get the following equivalent program to QAP.…”

Section: Quadratic Assignment Problemmentioning

confidence: 99%

Handbook of Semidefinite Programming

Wolkowicz¹,

Saigal²,

Vandenberghe³

2000

International Series in Operations Research &Amp; Management Science

463

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PrefaceSemide nite programming or SDP has been one of the most exciting and active research areas in optimization during the 1990s. It has attracted researchers with very diverse backgrounds, including experts in convex programming, linear algebra, numerical optimization, combinatorial optimization, control theory, and statistics. This tremendous research activity w as spurred by the discovery of important applications in combinatorial optimization and control theory, the development of e cient i n terior-point algorithms for solving SDP problems, and the depth and elegance of the underlying optimization theory.This book includes nineteen chapters on the theory, algorithms, and applications of semide nite programming. Written by the leading experts on the subject, it o ers an advanced and broad overview of the current state of the eld. The coverage is somewhat less comprehensive, and the overall level more advanced, than we had planned at the start of the project. In order to nish the book in a timely fashion, we h a ve had to abandon hopes for separate chapters on some important topics such as a discussion of SDP algorithms in the framework of general convex programming using the theory of self-concordant barriers, or an overview of numerical implementation aspects of interior-point methods for SDP. We apologize for any important gaps in the contents, and hope that the historical notes at the end of the book provide a useful guide to the literature on the topics that are not adequately covered in this handbook.We w ould like to thank all the authors for their outstanding contributions, their editorial help, and for their patience during the many revisions of the handbook. In addition, we thank Mike T odd for his valuable editorial advice on many occasions. We w ould also like to thank Erna Unrau for her help in combining some of the bibliographies into one. 11.5 -function, n K = 8 , n A = 10, = 1 0 ,5 , time limit one hour. 334 11.6 PLDI, n 500 where the variable is X 2 S n , the space of real symmetric n n matrices. The vector b 2 IR m , and the matrices A i 2 S n and C 2 S n are given problem parameters. The inequality X 0 means X is positive semide nite. More generally, throughout this book, the notation A B will mean B,A is positive semide nite. The notation C X stands for the inner product of the symmetric matrices C and X:Some authors write the inner product as C X = T r CXwhere Tr CXis the trace of the matrix CX. In other words, C X is a linear function of the elements X ij . In 1.1.1 we minimize a linear function of the matrix variable X, subject to m linear equality constraints A i X = b i , and the positive semide nitess constraint X 0. We refer to problem 1.1.1 as a semide nite program or SDP. W e can think of an SDP as a generalization of the standard form linear programming problem minimize c T x subject to a T i x = b i for all i = 1 ; : : : ; m x 0 1.1.2 1 2 HANDBOOK OF SEMIDEFINITE PROGRAMMING in which the elementwise nonnegativity constraint x 0 is replaced by a generalized inequality with re...

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