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Asymptotic estimations of a perturbed symmetric eigenproblem

Armand Gissler, Anne Auger, Nikolaus Hansen

Abstract

We study ill-conditioned positive definite matrices that are disturbed by the sum of $m$ rank-one matrices of a specific form. We provide estimates for the eigenvalues and eigenvectors. When the condition number of the initial matrix tends to infinity, we bound the values of the coordinates of the eigenvectors of the perturbed matrix. Equivalently, in the coordinate system where the initial matrix is diagonal, we bound the rate of convergence of coordinates that tend to zero.

Asymptotic estimations of a perturbed symmetric eigenproblem

Abstract

We study ill-conditioned positive definite matrices that are disturbed by the sum of rank-one matrices of a specific form. We provide estimates for the eigenvalues and eigenvectors. When the condition number of the initial matrix tends to infinity, we bound the values of the coordinates of the eigenvectors of the perturbed matrix. Equivalently, in the coordinate system where the initial matrix is diagonal, we bound the rate of convergence of coordinates that tend to zero.
Paper Structure (6 sections, 4 theorems, 20 equations, 1 figure)

This paper contains 6 sections, 4 theorems, 20 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Bounds on the eigenvalues of \ref{['eq:diagonal-plus-rankm']}
  3. Estimating the eigenvectors of \ref{['eq:diagonal-plus-rankm']}
  4. Rank-one perturbation
  5. Sum of $m$ rank-one matrices pertubation
  6. Thightness

Key Result

Theorem 1

If $e_i^{(m)}$ is a unit eigenvector corresponding to the $i$-th largest eigenvalue (counted with multiplicity) of $A^{(m)}$ in eq:diagonal-plus-rankm and $e_j^{(0)}$ is a unit eigenvector corresponding to the $j$-th largest eigenvalue $\lambda_j$ of $B$, then where $C_m>0$ is a constant which depends polynomially on $d$ and $\max_{k=1,\dots,m} \| v^{(k)}\|$.

Figures (1)

  • Figure 1: Value of $|[e_1^{(m)}]_j|$ as a function of $\lambda_1/\lambda_j$ where $e_1^{(m)}$ is an eigenvector associated to the largest eigenvalue of $A^{(m)}$ from \ref{['eq:diagonal-plus-rankm']} for different dimensions and values of $m$ as given in the legend. The $v^{(i)}$ are independent standard Gaussian vectors (with the same realization for all values of $\lambda_1$) and the eigenvalues of the diagional matrix $B$ are chosen uniformly on a $\log$ scale between $\lambda_d=1$ and $\lambda_1$. The value $|[e_1^{(m)}]_j|$ behaves consistent with $\Theta(\sqrt{\lambda_j/\lambda_1})$.

Theorems & Definitions (7)

  • Theorem 1
  • Lemma 2
  • proof
  • Proposition 3
  • proof
  • Theorem 4
  • proof