Tensor elds are at the heart of many science and engineering disciplines. Many tensor visualization methods separate the tensor into component eigenvectors and visualize those instead. Eigenvectors are normally ordered according to their eigenvalues: the eigenvectors corresponding to the smallest, median, or largest eigenvalues are in their corresponding groups. We showed that this ordering str...

Many eigenvalue problems are most naturally viewed as product eigenvalue problems. The eigenvalues of a matrix A are wanted, but A is not given explicitly. Instead it is presented as a product of several factors: A = AkAk−1 · · ·A1. Usually more accurate results are obtained by working with the factors rather than forming A explicitly. For example, if we want eigenvalues/vectors of BTB, it is b...

The purpose of this article is to improve existing lower bounds on the chromatic number χ. Let μ1, . . . , μn be the eigenvalues of the adjacency matrix sorted in non-increasing order. First, we prove the lower bound χ > 1 + maxm{ ∑m i=1 μi/ − ∑m i=1 μn−i+1} for m = 1, . . . , n − 1. This generalizes the Hoffman lower bound which only involves the maximum and minimum eigenvalues, i.e., the case...

The spinless Salpeter equation may be considered either as a standard approximation to the Bethe–Salpeter formalism, designed for the description of bound states within a relativistic quantum field theory, or as the most simple, to a certain extent relativistic generalization of the costumary nonrelativistic Schrödinger formalism. Because of the presence of the rather difficult-to-handle square...

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−1/2EA−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 qua...

The convergence of iterative methods used to solve the linear equations arising from radiosity systems depends on the distribution of the eigenvalues of the radiosity coefficient matrix. In this paper we prove that all eigenvalues of the radiosity coefficient matrix are real and positive. This fact may allow us to obtain fast radiosity solutions using the knowledge about the spectrum of the mat...

A probabilistic representation for a class of weighted p-radial distributions, based on mixtures cone probability measure and uniform distribution the Euclidean ℓpn-ball, is derived. Large deviation principles empirical coordinates random vectors ℓpn-ball with from this are discussed. The distributions extended to p-balls in classical matrix spaces, both self-adjoint non-self-adjoint matrices. ...

In this paper, we prove that the direction of cepstrum vectors strongly depends on vocal tract length and that this dependency is represented as rotation in a cepstrum space. In speech recognition studies, vocal tract length normalization (VTLN) techniques are widely used to cancel ageand gender-difference. In VTLN, a frequency warping is often carried out and it can be modeled as a linear tran...

Well known connections exist between the singular value decomposition of a matrix A and the Schur decomposition of its symmetric embedding sym(A) = ([ 0A ; A 0 ]). In particular, if σ is a singular value of A then +σ and −σ are eigenvalues of the symmetric embedding. The top and bottom halves of sym(A)’s eigenvectors are singular vectors for A. Power methods applied to A can be related to power...

This chapter is about eigenvalues and singular values of matrices. Computational algorithms and sensitivity to perturbations are both discussed. An eigenvalue and eigenvector of a square matrix A are a scalar λ and a nonzero vector x so that Ax = λx. A singular value and pair of singular vectors of a square or rectangular matrix A are a nonnegative scalar σ and two nonzero vectors u and v so th...