The paper addresses the problem of determining an outer interval solution of the parametric eigenvalue problem A(p)x = λx, A(p) ∈ ℝn×n for the general case where the matrix elements aij(p) are continuous nonlinear functions of the parameter vector p, p belonging to the interval vector p. A method for computing an interval enclosure of each eigenpair (λμ, x(μ)), μ = 1, ..., n, is suggested for the case where λμ is a simple eigenvalue. It is based on the use of an affine interval approximation of a
 ij
 (p) in p and reduces, essentially, to setting up and solving a real system of n or 2n incomplete quadratic equations for each real or complex eigenvalue, respectively.

Outer Interval Solution of the Eigenvalue Problem under General Form Parametric Dependencies

Computation of bounds for eigenvalues of structures with interval parameters

Computation of exact inertia and inclusions of eigenvalues (singular values) of tridiagonal (bidiagonal) matrices

In this article an explicit formula for eigenvalues of a 2-tridiagonal Toeplitz matrix can be derived on the basis of a certain relation between the determinant of this matrix and the determinant of a pertinent tridiagonal matrix. It can be pointed out that the problem is investigated without imposing any conditions on the elements of matrix.

/pdf/eigenvalues-of-2-tridiagonal-toeplitz-matrix-265jemfsdy.pdf

Eigenvalues of 2-tridiagonal Toeplitz matrix

This work is concerned with exploring the upper bounds and lower bounds of the eigenvalues of real symmetric matrices of order n whose entries are in a given interval. It gives the maximum and minimum of the eigenvalues and the upper bounds of spread of real symmetric interval matrices in all cases. It also gives the answers of the open problems for the maximum and minimum of the eigenvalues of real symmetric interval matrices. Copyright © 2012 John Wiley & Sons, Ltd.

Upper (lower) bounds of the eigenvalues, spread and the open problems for the real symmetric interval matrices

An algorithm for the determination of the eigenvalues of tridiagonal symmetric interval matrices is presented. The intervals of the eigenvalues are not overestimated, but exactly calculated. The algorithm is an efficient one, because it only needs twice as many operations as the Sturm algorithm, which is used for real matrices. >

Eigenvalues of tridiagonal symmetric interval matrices

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Eigenvalues of tridiagonal symmetric interval matrices