Fluid flows that are smooth at low speeds become unstable and then turbulent at higher speeds. This phenomenon has traditionally been investigated by linearizing the equations of flow and testing for unstable eigenvalues of the linearized problem, but the results of such investigations agree poorly in many cases with experiments. Nevertheless, linear effects play a central role in hydrodynamic ...

An equivariant matrix A commutes with a group of permutation matrices. Such matrices often arise in numerical applications where the computational domain exhibits geometrical symmetries, for instance triangles, cubes, or icosahedra. The theory for block diagonalizing equivariant matrices via the Generalized Fourier Transform (GFT) is reviewed and applied to eigenvalue computations. For dense ma...

The non-commuting graph $Gamma(G)$ of a non-abelian group $G$ with the center $Z(G)$ is a graph with thevertex set $V(Gamma(G))=Gsetminus Z(G)$ and two distinct vertices $x$ and $y$ are adjacent in $Gamma(G)$if and only if $xy neq yx$. The aim of this paper is to compute the spectra of some well-known NC-graphs.

Algebraic eigenvalues with associated left and right eigenvectors (nearly) orthogonal, and polynomial roots that are multiple, have been known to be sensitive to perturbations in numerical computation and thereby ill-conditioned. By constructing an extended polynomial system, it is proved that, under certain conditions, traditionally deened ill-conditioned eigenvalues and polynomial roots can b...

We present several variations on the harmonic Rayleigh–Ritz method. First, we introduce a relative harmonic approach for the standard, generalized, and polynomial eigenproblem. Second, a harmonic extraction method is studied for rightmost eigenvalues of generalized eigenvalue problems. Third, we propose harmonic extraction methods for large eigenvalues of generalized and polynomial eigenproblem...

We derive a new upper bound for the diameter of a k-regular graphG as a function of the eigenvalues of the adjacency matrix. Namely, supposethe adjacency matrix of G has eigenvalues AI , A2 .••.• An with lAd:::: IA21 ::::... :::: IAnl where AI = k, A = IA21. Then the diameter D(G) must satisfy D(G) :::; rlog(n 1)f1og(k/A)l. We wilJ consider families of graphs whose eigenvalues can b...

We study the relation between the diameter, the first positive eigenvalue of the discrete p-Laplacian and the `p-distortion of a finite graph. We prove an inequality relating these three quantities and apply it to families of Cayley and Schreier graphs. We also show that the `p-distortion of Pascal graphs, approximating the Sierpinski gasket, is bounded, which allows to obtain estimates for the...

Lemma 1.1. Let V be a finite-dimensional vector space over a field F. Let β, β′ be two bases for V . Let T : V → V be a linear transformation. Define Q := [IV ] ′ β . Then [T ] β β and [T ] ′ β′ satisfy the following relation [T ] ′ β′ = Q[T ] β βQ −1. Theorem 1.2. Let A be an n× n matrix. Then A is invertible if and only if det(A) 6= 0. Exercise 1.3. Let A be an n×n matrix with entries Aij, i,...

Mixing time of a Markov chain depends on the eigenvalues of its transition matrix. We give some examples and bounds on the mixing time in terms of the eigenvalue having second largest absolute value. This paper is based on Chapters 1, 4, and 12 of [1].

Most musical instruments are built from physical systems that oscillate at certain natural frequencies. The frequencies are the imaginary parts of the eigenvalues of a linear operator, and the decay rates are the negatives of the real parts, so it ought to be possible to give an approximate idea of the sound of a musical instrument by a single plot of points in the complex plane. Nevertheless, ...