2 edition of **Bounds for the eigenvalues of a matrix** found in the catalog.

Bounds for the eigenvalues of a matrix

Kenneth R. Garren

- 21 Want to read
- 6 Currently reading

Published
**1968** by National Aeronautics and Space Administration; for sale by the Clearinghouse for the Federal Scientific and Technical Information, Springfield, Va. in [Washington] .

Written in English

- Matrices.,
- Eigenvalues.

**Edition Notes**

Statement | by Kenneth R. Garren. |

Series | NASA technical note, NASA TN D-4373, NASA technical note ;, D-4373. |

Classifications | |
---|---|

LC Classifications | TL521 .A3525 no. 4373 |

The Physical Object | |

Pagination | 44 p. |

Number of Pages | 44 |

ID Numbers | |

Open Library | OL5634691M |

LC Control Number | 68061717 |

The classical Chernoff bounds concern the sum of independent, nonnegative, and uniformly bounded random variables. In the matrix setting, the analogous theorem concerns a sum of positive-semidefinite random matrices subjected to a uniform eigenvalue bound. Matrix Chernoff I. Review of Perturbation Bounds for Matrix Eigenvalues by Ft. Bhatia Raphael Loewy Department of Mathematics Technion - lsrael institute of Technology obtaining information about the variation of the eigenvalues of a matrix under small perturbations, which is central in perturbation theory, is a special case of BOOK REVIEW Let A, BEN. vertices. As a result the best known lower bounds for the chromatic number are spectral [19], and in this paper we improve these bounds by incorporating all eigenvalues. We also conjecture a relationship between the sign of the eigenvalues and the chromatic number, which if true could lead to further developments in spectral graph by: for eigenvalues of Hermitian matrices when the perturbation is Hermitian of rank one. Before describing an application that requires such bounds, we mention that algorithms for computing eigenvalues and eigenvectorsof Hermitian matrices modiﬁed by a rank one matrix are well established [3, 10, 14], [11, §, §]; and that the.

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Perturbation Bounds for Matrix Eigenvalues contains a unified exposition of spectral variation inequalities for matrices. The text provides Bounds for the eigenvalues of a matrix book complete and self-contained collection of bounds for the distance between the eigenvalues of two matrices, which could be Cited by: Bounds for the eigenvalues of a matrix.

[Washington] National Aeronautics and Space Administration; for sale by the Clearinghouse for the Federal Scientific and Technical Information, Springfield, Va.

[] (OCoLC) Document Type: Book: All Authors /. Perturbation Bounds for Matrix Eigenvalues contains a unified exposition of spectral variation inequalities for matrices. The text Bounds for the eigenvalues of a matrix book a complete and self-contained collection of bounds for the distance between the eigenvalues of two matrices, which could be.

Bounds for the smallest and the largest eigenvalues of Hermitian matrices 2 The main result Lemma Let A= [a ij] be an n nHermitian matrix.

Then the eigenvalues of a given n nHermitian matrix A= [a ij] other than 1(A) and n(A), we can derive an upper Bounds for the eigenvalues of a matrix book that is dependent on the entries of AandFile Size: KB.

We obtain bounds for the largest and least eigenvalues of the adjacency matrix of a simple undirected graph. We find upper bound for the second largest eigenvalue of the adjacency matrix. We prove that the bounds obtained here improve on the existing bounds Author: Pintu Bhunia, Santanu Bag, Kallol Paul.

Novel bounds are proposed for the extreme and lower half eigenvalues of the solution matrix for the algebraic Riccati equation. The formulae giving these bounds can easily be applied to determine the region where the eigenvalues lie, and the bounds have the added advantage of being sharper in some cases than the previously proposed ones, as some realistic examples will by: 8.

Upper and lower bounds are derived for the absolute values of the eigenvalues of a matrix polynomial (or λ-matrix). The bounds are based on norms of the coefﬁcient matrices and involve the inverses of the leading and trailing coefﬁcient matrices.

ERROR BOUNDS FOR THE EIGENVALUES OF THE KERNEL MATRIX the underlying principles which permit the derivation of scaling bounds for individual eigenvalues ﬁrst. In a second step, one could then use these results to construct statistical tests to estimate, for.

Eigenvalue variance bounds for covariance matrices S. Dallaporta University of Toulouse and CMLA, ENS Cachan Bounds for the eigenvalues of a matrix book. This work is concerned with ﬁnite range bounds on the variance of individual eigenvalues of random covariance matrices, both in the bulk and at the edge of the spectrum.

In a preceding paper, the author established analogousFile Size: KB. Yes, it is true that the largest eigenvalue is bounded by the largest absolute row sum or column sum. You can check Gershgorin circle theorem. Actually, all the eigenvalues lie in. The first version of this book was written in and published in Its principal goal was the exposition of bounds for the distance between the eigenvalues of two matrices Aand Bin terms of expressions involving ‖A− B‖.

The prototype of such bounds is H. Weyl's inequality from I have bounds on both the minimum and Stack Exchange Network Stack Exchange network consists of Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

The eigenvalues of the kernel matrix play an important role in a number of kernel methods, in particular, in kernel principal component analysis. It is well known that the eigenvalues of the kernel matrix converge as the number of samples tends to Bounds for the eigenvalues of a matrix book L BraunMikio.

The resulting improvement is more important in the matrix case, since computing eigenvalues of matrix polynomials is significantly more difficult than computing polynomial zeros.

Among others, bounds on such eigenvalues are useful in their computation by iterative methods and when computing pseudospectra. We first illustrate our result by Cited by: 7.

For two Hermitian matrices A and B, at least one of which is positive semidefinite, we give upper and lower bounds for each eigenvalue of AB in terms of the eigenvalues of A and B. For two complex matrices A,B with known singular values, upper and lower bounds are deduced for each singular value of by: On lower bounds for the largest eigenvalue of a symmetric matrix S.G.

Walker a,∗, P. Van Mieghem b a Institute of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury CT2 7NZ, UK b Delft University of Technology, Faculty of Electrical Engineering, Mathematics and Computer Science, P.O. BoxGA Delft, The. Perturbation Bounds for Matrix Eigenvalues contains a unified exposition of spectral variation inequalities for matrices.

The text provides a complete and self-contained collection of bounds for the distance between the eigenvalues of two matrices, which could.

In this paper, an upper bound on the spectral radius ρ (A ∘ B) for the Hadamard product of two nonnegative matrices (A and B) and the minimum eigenvalue τ (C ★ D) of the Fan product of two M-matrices (C and D) are researched.

These bounds complement some corresponding results on the simple type bounds. In addition, a new lower bound on the minimum eigenvalue of the Fan product Author: Qianping Guo, Jinsong Leng, Houbiao Li, Carlo Cattani.

We will call a matrix positive matrix if all elements in the matrix are positive, and we will denote the largest eigenvalue with $\lambda_{\max}$, what is exist because of the Perron–Frobenius theorem. To study the lower bound for the minimum eigenvalue and a upper bound for the spectral radius of Hadamard product of two irreducible M-matrices A and B, obtaining some new estimation of the bounds.

These new bounds are only depend on the element of A and B, so they are easy to : Jiaozhen Zhao. For more general graphs, perturbation theory for matrix eigenvalues can provide upper bounds on the difference between the eigenvalues of K and K + δK, but these bounds are too conservative, Author: Rajendra Bhatia.

Bounds on eigenvalues of matrix products with an application to the algebraic Riccati equation Abstract: Lower and upper summation bounds for the eigenvalues of the product XY are presented, under various restrictions on matrices X, Y in R/sup n*n/.

An application to the algebraic Riccati equation yields a trace lower by: Stack Exchange network consists of Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share.

4 On simple bounds for eigenvalues Furthermore, (5) with S= {3}and S= {2,4}, also gives λ 1 ≤0,λ 2 ≤1.

The key aspect we want to stress out is that all these bounds are File Size: KB. In mathematics, an eigenvalue perturbation problem is that of finding the eigenvectors and eigenvalues of a system that is perturbed from one with known eigenvectors and eigenvalues. This is useful for studying how sensitive the original system's eigenvectors and eigenvalues are to changes in the system.

This type of analysis was popularized by Lord Rayleigh, in his investigation of harmonic. The bounds are found by independent calculations on the elements, and, for certain element types, are independent of mesh irregularity. We give examples of when some of the bounds are attained. These results indicate that the preconditioned conjugate-gradient method is appropriate and very rapid for the solution of Galerkin mass-matrix by: Matrix Concentration Bounds In applications, it is common that a random matrix can be expressed as a sum of independent random matrices[1].

For example, the covariance of X 2 Rn⇥d can be written as XTX = Pn i=1 x T i xi where xi denotes i-th row of X. In this section, we state two common bounds on random matrices[1]. Matrix Chernoff. Upper bounds for the eigenvalue of S k Proof It is not difﬁcult to prov e that, if B 1 is the unit ball centered at the origin, we have t + ρ B 1 ⊂ t − ρ, 0.

Summary: "Perturbation Bounds for Matrix Eigenvalues contains a unified exposition of spectral variation inequalities for matrices.

The text provides a complete and self-contained collection of bounds for the distance between the eigenvalues of two matrices. Moreover, Yan derived from the Courant–Weyl inequalities the relation (2) 2 λ 1 (G) ≤ q 1 (G) where λ 1 is the index of the adjacency matrix of G.

These relations immediately imply that any lower bound on μ 1 (G) is a valid lower bound on q 1 (G) and that doubling any lower bound on λ 1 (G) also yields a valid lower bound on q 1 (G).Cited by: texts All Books All Texts latest This Just In Smithsonian Libraries FEDLINK Lower bounds for the rank and location of the Eigenvalues of a matrix Item Preview Lower bounds for the rank and location of the Eigenvalues of a matrix by Fan, Ky; Hoffman, A.

Topics Eigenvalues, Matrices Publisher National Bureau of Standards. New bounds on the minimum eigenvalue of matrices (ii) If Ais a strictly diagonally dominant matrix by columns, then 1 a ii v ii 1 a ii+ Pn j=1;j6=i ij j 1 C.

CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): Abstract. In a recent paper Melman [12] derived upper bounds for the smallest eigenvalue of a real symmetric Toeplitz matrix in terms of the smallest roots of rational and polynomial approximations of the secular equation f(λ) =0, the best of which being constructed by the (1, 2)-Padé approximation of f.

This work introduces the minimax Laplace transform method, a modification of the cumulant-based matrix Laplace transform method developed in "User-friendly tail bounds for sums of random matrices" (arXivv6) that yields both upper and lower bounds on each eigenvalue of a sum of random self-adjoint matrices.

The eigenvalues of a matrix are not easy to compute. It is remarkable, therefore, that with relatively simple mental arithmetic, you can obtain bounds for the eigenvalues of a matrix of any size.

The bounds are provided by using a marvelous mathematical result known as Gershgorin's Disc Theorem. For certain. In addition, we found the bounds for k-th eigenvalues of normalized Laplacian matrix and signless Laplaci\ an matrix. Keywords: Normalized Laplacian Matrix, Signless Laplacian Matrix, Bounds of Eigenvalue Created Date: 11/14/ PMFile Size: 2MB.

Let A S n [a,b] denote a set of all real nxn symmetric matrices with entries in the interval [a,b].In this article, we present bounds for the second largest eigenvalue λ 2 (A) of a real symmetric matrix A, such that A∈A S 3 [-b,b].Cited by: 2.

inde nite linear systems, eigenvalues, condition number, inertia, eigenvalue bounds, regularization 1. Introduction. Given a symmetric and positive semide nite Hessian matrix H2R n, vectors c2Rn and b2Rm, and a Jacobian matrix J2Rm n, where m n, consider the primal-dual pair of quadratic programs (QP) in standard form minimize x cTx+ 1 2 x.

singular then 0is an eigenvalue of the matrix polynomial zmP(1/z). There-fore, to locate the eigenvalues of these matrix polynomials, we always assume that A 0 and Am are non-singular. The paper is organized as follows.

In Section 2 we give bounds for matrix polynomials whose coefficients satisfy some special properties, in particular. Summary. This book deals with the determinants of linear operators in Euclidean, Hilbert and Banach spaces. Determinants of operators give us an important tool for solving linear equations and invertibility conditions for linear operators, enable us to describe the spectra, to evaluate the multiplicities of eigenvalues, etc.

We derive upper and lower bounds on the smallest and pdf eigenvalues, respectively, of real symmetric Toeplitz matrices. The bounds are first obtained for positive-definite matrices and then extended to the general real symmetric case.

Perturbation Bounds for Matrix Eigenvalues contains a unified exposition of spectral variation inequalities for matrices. The text provides a complete and self-contained collection of bounds for the distance between the eigenvalues of two matrices, which could be .We ebook characterize the extremal graphs attaining these bounds.

As an special cases of our results, we ebook be able to cover some of the bounds recently given in the literature for the case of distance matrix and distance signless Laplacian matrix. We also obtain new bounds for the k-th generalized distance by: 2.