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1 edition of Bounds on the extreme generalized eigenvalues of Hermitian pencils found in the catalog.

Bounds on the extreme generalized eigenvalues of Hermitian pencils

Monique P. Fargues

# Bounds on the extreme generalized eigenvalues of Hermitian pencils

## by Monique P. Fargues

Published by Naval Postgraduate School, Available from National Technical Information Service in Monterey, Calif, Springfield, Va .
Written in English

Subjects:
• Eigenvalues

We present easily computable bounds on the extreme generalized eigenvalues of Hermitian pencils (R,B) with finite eigenvalues and positive definite B matrices. The bounds are derived in terms of the generalized eigenvalues of the subpencil of maximum dimension contained in (R,B). Known results based on the generalization of the Gershgorin theorem and norm inequalities are presented and compared to the proposed bounds. It is shown that the new bounds compare favorably with these known results; they are easier to compute, require less restrictions on the properties of the pencils studied, and they are in an average sense tighter than those obtained with the norm inequality bounds.

Edition Notes

The Physical Object ID Numbers Other titles NPS-62-90-016. Statement by Monique P. Fargues Contributions Naval Postgraduate School (U.S.). Dept. of Electrical and Computer Engineering Pagination 21 p. : Number of Pages 21 Open Library OL25495719M

Numerical Optimization of Eigenvalues of Hermitian Matrix Functions Mustafa K l ˘c Emre Mengiy E. Alper Y ld r mz Febru Abstract The eigenvalues of a Hermitian matrix function that depends on one parameter analyt-ically can be ordered so that each eigenvalue is .   We prove that eigenvalues of a Hermitian matrix are real numbers. This is a finial exam problem of linear algebra at the Ohio State University. Two proofs given. We prove that eigenvalues of a Hermitian matrix are real numbers. This is a finial exam problem of .

Hermitian Operators •Definition: an operator is said to be Hermitian if –The degree of degeneracy’ of an eigenvalue is the number of linearly independent eigenvectors that are associated with it lower bounds can be either finite or infinite •The number of basis vectors is uncountable infinity’. Upper bounds on eigenvalues of PSD matrix? Ask Question Asked 7 years, 11 months ago. Active 2 years, 9 months ago. Viewed 9k times 6. 5 $\begingroup$ Suppose A is a symmetric positive semidefinite matrix. Is there a way to upper bound the largest eigenvalue using properties of its row sums or column sums? For instance.

Chapter & Page: 7–2 Eigenvectors and Hermitian Operators! Example Let V be the vector space of all inﬁnitely-differentiable functions, and let be the differential operator (f) = f ′′.Observe that (sin(2πx)) = d2 dx2 sin(2πx) = −4π2 sin(2πx). Thus, for this operator, −4π2 is an eigenvalue with corresponding eigenvector sin(2πx) Stack Exchange network consists of Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share .

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### Bounds on the extreme generalized eigenvalues of Hermitian pencils by Monique P. Fargues Download PDF EPUB FB2

Title from cover "NPS" "December " AD A Includes bibliographical references (p. 14) We present easily computable bounds on the extreme generalized eigenvalues of Hermitian pencils (R,B) with finite eigenvalues and positive definite B :   texts All Books All Texts latest This Just In Smithsonian Libraries FEDLINK Bounds on the extreme generalized eigenvalues of Hermitian pencils Bounds on the extreme generalized eigenvalues of Hermitian pencils by Fargues, Monique P.

Publication date TZ Topics EIGENVALUES. S^5 NPS NAVALPOSTGRADUATESCHOOL Monterey,California BOUNDSONTHEEXTREMEGENERALIZED EIGENVALUESOFHERMITIANPENCILS by s December FedDocs D/2 NPS Approvedforpublicrelease;distributionisunlimited Preparedfor:NavalPostgraduateSchool.

We present easily computable bounds on the extreme generalized eigenvalues of Hermitian pencils (R,B) with finite eigenvalues and positive definite B matrices.

The bounds are derived in terms of the generalized eigenvalues of the subpencil of maximum dimension contained in (R,B).Author: Monique P. Fargues. Weyl-type eigenvalue perturbation theories are derived for Hermitian definite pencils A-λB, in which B is positive definite.

The results provide a one-to-one correspondence between the original Author: Chen Xiao Shan. This paper is concerned with the Hermitian definite generalized eigenvalue problem A-λ B for block diagonal matrices A = diag (A 11, A 22) and B = diag (B 11, B 22).Both A and B are Hermitian, and B is positive definite.

Bounds on how its eigenvalues vary when A and B are perturbed by Hermitian matrices are established. These bounds are generally of linear order with respect to the. A Minimax Characterization for Eigenvalues of Hermitian Pencils B.

Najman* and Q. Yet Department of Mathematics and Statistics University of Calgary Calgary, Alberta T2N 1N4, Canada Submitted by Peter Lancaster ABSTRACT We establish a minimax characterization for extreme real eigenvalues of a general hermitian matrix pencil.

To illustrate the idea, suppose that B has only 1 negative eigenvalue and n − 1 positive eigenvalues. Then, our results show that at least n − 2 eigenvalues are real; see the left plot of Fig.

2 for an illustration with n = inertia of A is (3, 0, 3), so three of the six curves λ i (A − t B) take positive values at t = t → ∞, however, the inertia of A − t B must match. The following corollary of Theorem gives a lower bound for the spread of a Hermitian matrix in terms of its entries.

Corollary Let A= [a ij] be an n nHermitian matrix, with n 2. Then Bounds for the smallest and the largest eigenvalues of Hermitian matrices Corollary Let Aand Cbe two Hermitian matrices that are similar to. Perturbation analysis for the generalized eigenvalue and the generalized singular value problem.

Matrix Pencils, () Perturbation thèorems for the generalized eigenvalue. Let an n × n Hermitian matrix A be presented in block 2 × 2 form as $$A = \left[ {\begin{array}{*{20}c} {A_{11} } & {A_{12} } \\ {A_{12}^* } & {A_{22} } \\ \end{array} } \right]$$, where A 12 ≠ 0, and assume that the diagonal blocks A 11 and A 22 are positive definite.

Under these assumptions, it is proved that the extreme eigenvalues of A satisfy the bounds. CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): In this note, we study some basic properties of generalized eigenvalues of a definite Hermitian matrix pair.

In particular, we prove an interlacing theorem and a minimax theorem. We also obtain upper bounds for the variation of the generalized eigenvalues under perturbation.

L():= EA is an even matrix pencil, that is, L() = L(). The eigenvalues of L() are symmetric wrt the imaginary axis. L(i) = i EA is a Hermitian matrix pencil. The eigenvalues of L(i) are symmetric wrt the real axis.

If Ris invertible, then the generalized eigenvalue problem can be reduced to the form 0 E E 0 Q SR 1S A SR 1B A BR 1S BR 1B.

This paper considers the sensitivity of semisimple multiple eigenvalues and corresponding generalized eigenvector matrices of a nonsymmetric matrix pencil analytically dependent on several parameters. Let $$A,B \in\mathbb{C}^{n\times n}$$ be Hermitian matrices with B being positive definite.

We now consider a perturbation problem for $$A\boldsymbol{x}= \lambda B\boldsymbol{x}$$.It is known that the n generalized eigenvalues of the matrix pencil $$\langle A,B\rangle$$ are real numbers and that the generalized eigenvalues of $$\langle A,B\rangle$$ and the eigenvalues of $$AB^{-1}$$ are the same.

Abstract. Perturbation bounds for the generalized eigenvalue problem of a di-agonalizable matrix pencil A-ÀB with real spectrum are developed.

It is shown how the chordal distances between the generalized eigenvalues and the angular distances between the generalized eigenspaces can be bounded in terms of the angular distances between the matrices.

[2] Amir-Moéz, A. R., ‘ Extreme properties of eigenvalues of a Hermitian transformation and singular values of the sum and product of linear transformations ’, Duke Math. Bounds on the Extreme Generalized Eigenvalues of Hermitian Pencils 12 PEPSONAL AUTHOR(S) Monique P.

Fargues 13a TYPE OF REPORT 13b TIME COVERED /14 DATE OF REPORT (Year, Month Day) 15 PAGE COUNT Technical Report FROM 7/1/90 TO 12/31/ 0 Decmeber 31 20 16 SUPPLEMENTARY NOTATION. Perturbation bounds for the generalized Schur decomposition SIAM J. Matrix Anal.

Appl., 16, () A note on backward perturbations for the Hermitian eigenvalue problem BIT, 35, () Optimal backward perturbation bounds for the linear LS problem with multiple right-hand sides IMA J. Numer. Anal., 16, ().

Lower bounds of the minimal eigenvalue of a Hermitian positive-definite matrix Abstract: In this correspondence, we present several lower bounds of the minimal eigenvalue of a class of Hermitian positive-definite matrices, which improve the previous bounds given by Dembo () and Ma and Zarowski ().

Chapter 5 is dedicated to the Generalized Eigenvalue Problem (GEP). The essence of the chapter is concerned with the techniques used in computing the eigenvalues and eigenvectors of Hermitian pencils.

Special emphasis is put on.The Perturbation of Eigenvalues. Invariant Subspaces. Generalized Eigenvalue Problems. R. Horn and C.

Johnson (). Matrix Analysis, Cambridge University Press, New York. Review and Miscellanea. Eigenvalues, Eigenvectors, and Similarity. Unitary Equivalence and Normal Matrices.

Canonical Forms. Hermitian and Symmetric Matrices.We consider matrix pencils λA — B in which λ is a complex parameter, A, B are both hermitian and A is nonsingular. Variational characterizations of the real eigenvalues (if any) are formulated.

Rayleigh quotient algorithms for finding real eigenvalues are proposed and their local and global convergence properties are established and.