Matrix Perturbation Theory in the Tangent Space of Isospectral Matrices

Abstract

Eigenvalue and eigenvector perturbation theory is a fundamental topic in several disciplines, including numerical linear algebra, quantum physics, and related fields. The central problem is to understand how the eigenvalues and eigenvectors of a matrix $A \in \mathbb{C}^{n \times n}$ change under the addition of a perturbation matrix $E \in \mathbb{C}^{n \times n}$. Much of the existing literature focuses on structured perturbations. For example, in [C.-K. Li and R.-C. Li, Linear Algebra Appl. 2005], the matrix $A$ is assumed to be Hermitian and block diagonal, while the perturbation $E$ is Hermitian and block off-diagonal. In this work, we investigate a different structured setting in which the perturbation has the commutator form $E = AB - BA$ for some matrix $B$, which we show to be a generalization of the block diagonal structure considered by Li and Li. First, we extend their main result by showing that the perturbation of the $i$-th eigenvalue of $A$, denoted by $λ_i$, is of order $\|E\|^2 / η_i$, where $η_i = \min_{j \neq i} |λ_i - λ_j|$ is the spectral gap associated with $λ_i$. Second, we provide a detailed analysis of the role played by the matrix $B$ in the perturbation of the eigenvectors. This analysis is further generalized to the case of block-diagonal matrices with multiple eigenvalues, as well as to perturbed singular values and eigenvalues of Jordan blocks.

Figures (6)

• Figure 1: Plot of bounds in \ref{['thm:boundeigenvec']}. The red dots are the actual norm of the residuals, the blue line is the bound when we have used $\|B(:,i)\| \le \|E\|/\eta_i$ and the brown circles are the bound where $\|B(:,i)\|$ is kept as it is. $|\Delta(t)|$ in is bounded by \ref{['thm:boundeige2']}. The matrix considered is a diagonal matrix with eigenvalues equispaced from $1$ to $200$ and $B$ is a random off-diagonal matrix such that $\|E\| = 10^{-3}$. Note that $\rho_i$ is constant and equal to $10^{-3}$. The blue line in the linear bound is $\approx \|E\|/\eta_i$, which is the classical bound presented for perturbed eigenvectors.
• Figure 1: Bound in \ref{['thm:blkvecherm']} where $D \in \mathbb{R}^{200,200}$, $\lambda_1 = 1$ and $D_2$ is diagonal with eigenvalues equispaced from $2$ to $101$. The matrix $B_1$ is random and scaled such that $\|E\| = 10^{-2}$. Note that this bound is much tighter than $(\|E\|/gap)^3 = 10^{-6}$.
• Figure 1: Netwon diagram in the generic case where the coefficient of $\varepsilon^2t^0$ is not zero. The slope of the solid line indicates the power of $\varepsilon$ in the perturbation of the eigenvalues ($2/n$).
• Figure 2: Plot of bounds in \ref{['thm:boundeigenvec']}. The red dots are the actual norm of the residual, the blue line is the bound when we have used $\|B(:,i)\| \le \|E\|/\eta_i$ and the brown circles are the bound where $\|B(:,i)\|$ is kept as it is. $|\Delta(t)|$ in is bounded by \ref{['thm:boundeige2']}. The matrix considered is a diagonal matrix with eigenvalues equal to the square root of the equispaced points from $1$ to $200$ and $B$ is a random off-diagonal matrix such that $\|E\| = 10^{-3}$. Note that, for the largest eigenvalues, $\rho_i \approx 10^{-1}$. The blue line in the linear bound is $\approx \|E\|/\eta_i$, which is the classical bound presented for perturbed eigenvectors.
• Figure 2: Perturbed eigenvalues for a nilpotent Jordan block of size $100 \times 100$ and a perturbation $E$ in the tangent with norm $\|E\| = 10^{-3}$. The circle has radius $\|E\|^{2/100}$.

Theorems & Definitions (25)

• Theorem 2.1: Another characterization of the tangent space
• Lemma 3.1
• Proof 1
• Lemma 3.2
• Lemma 3.3
• Definition 3.4
• Lemma 3.5
• Proof 2
• Lemma 3.6
• Proof 3