In this article we further develop a perturbation approach to the Rayleigh– Ritz approximations from our earlier work. We both sharpen the estimates and extend the applicability of the theory to nonnegative definite operators . The perturbation argument enables us to solve two problems in one go: We determine which part of the spectrum of the operator is being approximated by the Ritz values an...

Applications related to artificial intelligence, machine learning, and system identification simulations essentially use eigenvectors. Calculating eigenvectors for very large matrices using conventional methods is compute-intensive renders the applications slow. Recently, Eigenvector-Eigenvalue Identity formula promising significant speedup was identified. We study algorithmic implementation of...

This paper addresses the problem of direction-of-arrival (DOA) estimation of correlated and coherent signals, and two sparsityinducing methods are proposed. In the first method named L1-EVD, the signal-subspace eigenvectors are represented jointly with wellchosen hard thresholds attached to the representation residue of each eigenvector. Then only the eigenvector corresponding to the largest ei...

This paper introduces a simple variant of the power method. It is shown analytically and numerically to accelerate convergence dominant eigenvalue/eigenvector pair, it particularly effective for problems featuring small spectral gap. The introduced method one-step extrapolation technique that uses linear combination current previous update steps form better approximation eigenvector. provided a...

Let A = (aij ) ∈ Rn×n,N = {1, . . . , n} and DA be the digraph (N, {(i, j); aij > −∞}). The matrix A is called irreducible if DA is strongly connected, and strongly irreducible if every maxalgebraic power of A is irreducible. A is called robust if for every x with at least one finite component, A(k) ⊗ x is an eigenvector of A for some natural number k. We study the eigenvalue–eigenvector proble...

1.1.6 Let p(•) be any polynomial. If λ ∈ σ(A) with associated eigenvector y, then p(λ) ∈ σ(p(A)) also with associated eigenvector y. 1.1.p1 λ ∈ σ(A)⇔ λ−1 ∈ σ(A−1). 1.2.4-e If T ∈Mn is triangular, then σ(A) = {t11, ..., tnn}. 1.2.6-e If A ∈Mn(R) and n is odd, A has at least one real eigenvalue. 1.2.5 Ek(A) is the sum of the principle minors (the square k × k submatrices lying along the diagonal)...

We investigate eigenvector statistics of the Truncated Unitary ensemble TUE(N,M) in weakly non-unitary case M=1, that is when only one row and column are removed. provide an explicit description generalized overlaps as deterministic functions eigenvalues, well a method to derive exact formula for expectation diagonal (or squared eigenvalue condition numbers), conditionally on eigenvalue. This c...

A theoretical framework is presented for understanding the effects of noise on estimates of the eigenvalues and eigenvectors of the diffusion tensor at moderate to high signal-to-noise ratios. Image noise produces a random perturbation of the diffusion tensor. Power series solutions to the eigenvalue equation are used to evaluate the effects of the perturbation to second order. It is shown that...