Improving eigenvalue bounds using extra bounds (original) (raw)

On relative residual bounds for the eigenvalues of a Hermitian matrix

1996

Let H be a Hermitian matrix, X an orthonormal matrix, and M = X*HX. Then the eigenvalues of M approximate some eigenvalues of H with an absolute error bounded by I]ttX-XMI]2. The main interest in this work is the relative distance between the eigenvalues of M and some part of the spectrum of H. It is shown that distance depends on the angle between the ranges of X and HX.

On inequalities involving eigenvalues and traces of Hermitian matrices

Annals of Functional Analysis, 2015

It is shown that some immediate consequences of the spectral theorem provide refinements and extensions of the several well-known inequalities involving eigenvalues and traces of Hermitian matrices. We obtain bounds for the spread and condition number of a Hermitian matrix.

Bounds for the smallest and the largest eigenvalues of Hermitian matrices

International Journal of Algebra, 2015

In this paper the author derives a lower bound for the largest eigenvalue and an upper bound for the smallest eigenvalue of Hermitian matrices, based on Weyl's inequalities. Some related results, consequences, applications, and examples are provided. In the fifth section, the main result is applied to some integer matrices.

Bounds for eigenvalues using the trace and determinant

Linear Algebra and its Applications, 1997

Let A be a square matrix with real and positive eigenvalues A 1 >/ --->1 A n > 0, and let 1 ~< k ~< I ~< n. Bounds for A k "-A l and A k + "" + A t, involving k, 1, n, tr A, and det A only, are presented. @ 1997 Elsevier Seienee Inc.

Bounds for sums of eigenvalues and applications

Computers & Mathematics with Applications, 2000

Let A be a matrix of order n × n with real spectrum A 1 ~ A2 ~_ " • • _~> An. Let 1 < k < n -2. If An or A1 is known, then we find an upper bound (respectively, lower bound) for the sum of the k-largest (respectively, k-smallest) remaining eigenwlues of A. Then, we obtain a majorization vector for (A1, A2,..., An-l) when An is known and a majorization vector for (A2, A3,..., An) when A1 is known. We apply these results to the eigenvalues of the Laplacian matrix of a graph and, in particular, a sufficient condition for a graph to be connected is given. Also, we derive an upper bound for the coefficient of ergodicity of a nonnegative matrix with real spectrum. (~)

Weyl-type relative perturbation bounds for eigensystems of Hermitian matrices

Linear Algebra and its Applications, 2000

We present a Weyl-type relative bound for eigenvalues of Hermitian perturbations A + E of (not necessarily definite) Hermitian matrices A. This bound, given in function of the quantity η = A −1/2 EA −1/2 2 , that was already known in the definite case, is shown to be valid as well in the indefinite case. We also extend to the indefinite case relative eigenvector bounds which depend on the same quantity η. As a consequence, new relative perturbation bounds for singular values and vectors are also obtained. Using matrix differential calculus techniques we obtain for eigenvalues a sharper, first-order bound involving the logarithm matrix function, which is smaller than η not only for small E, as expected, but for any perturbation.

Diagonals and Eigenvalues of Sums of Hermitian Matrices: Extreme Cases

Proyecciones, 2003

There are well known inequalities for Hermitian matrices A and B that relate the diagonal entries of A+B to the eigenvalues of A and B. These inequalities are easily extended to more general inequalities in the case where the matrices A and B are perturbed through congruences of the form U AU * + V BV * , where U and V are arbitrary unitary matrices, or to sums of more than two matrices. The extremal cases where these inequalities and some generalizations become equalities are examined here.