When is the Resolvent Like a Rank One Matrix?

TL;DR

This work studies when the resolvent $ (A-zI)^{-1}$ is well approximated by a rank-one matrix by introducing the set $S_{\epsilon}(A)$ defined via the ratio of the smallest two singular values of $A-zI$, and relating it to the $\epsilon$-pseudospectrum $\Lambda_{\epsilon}(A)$. It develops a framework tying $S_{\epsilon}(A)$ to pseudospectra through bounds and disks around eigenvalues, with concrete results for Jordan blocks and large Toeplitz matrices via the splitting theorem, as well as a general perturbation analysis that highlights the role of near-orthogonality of left/right singular vectors. Key contributions include quantitative inclusions between $\Lambda_{\epsilon}(A)$ and $S_{\epsilon}(A)$, explicit eigenvalue-centered disks contained in $S_{\epsilon}(A)$ for certain matrix classes, and a differentiation theory for singular values and vectors that links their evolution to the rank-one approximation property. The findings illuminate when large resolvents admit low-rank representations, enabling cheaper computations in applications such as fluid mechanics, and raise open questions about the matrix classes and regions where these rank-one approximations remain valid.

Abstract

For a square matrix $A$, the resolvent of $A$ at a point $z \in \mathbb{C}$ is defined as $(A-zI )^{-1}$. We consider the set of points $z \in \mathbb{C}$ where the relative difference in 2-norm between the resolvent and the nearest rank one matrix is less than a given number $ε\in (0,1)$. We establish a relationship between this set and the $ε$-pseudospectrum of $A$, and we derive specific results about this set for Jordan blocks and for a class of large Toeplitz matrices. We also derive disks about the eigenvalues of $A$ that are contained in this set, and this leads to some new results on disks about the eigenvalues that are contained in the $ε$-pseudospectrum of $A$. In addition, we consider the set of points $z \in \mathbb{C}$ where the absolute value of the inner product of the left and right singular vectors corresponding to the largest singular value of the resolvent is less than $ε$. We demonstrate numerically that this set can be almost as large as the one where the relative difference between the resolvent and the nearest rank one matrix is less than $ε$ and we give a partial explanation for this. Some possible applications are discussed.

Figures (10)

• Figure 1: Contour plots of ratios of second largest to largest singular value of the resolvent (left) and of absolute value of inner product of left and right singular vectors corresponding to largest singular value of the resolvent (right). Matrix is the Grcar matrix of size $n=50$. Eigenvalues are shown with red asterisks.
• Figure 2: Contour plots of ratios of second largest to largest singular value of the resolvent (left) and of absolute value of inner product of left and right singular vectors corresponding to largest singular value of the resolvent (right). Matrix is the transient_demo matrix of size $n=50$. Eigenvalues are shown with red asterisks.
• Figure 3: Contour plots of ratios of second largest to largest singular value of the resolvent (left) and of absolute value of inner product of left and right singular vectors corresponding to largest singular value of the resolvent (right). Matrix is the sampling matrix of size $n=10$. Eigenvalues are shown with red asterisks.
• Figure 4: Contour plot of ratios of second largest to largest singular value of the resolvent for a normal matrix with the same eigenvalues as the Grcar matrix of size $n=50$. The $0.5$ contour is plotted because contours with much smaller values are barely visible on the scale of the graph. Eigenvalues are shown with red asterisks.
• Figure 5: Comparison of curve where $\sigma_2 ( (A-zI )^{-1}) / \sigma_1 ( (A-zI )^{-1}) = 10^{-3}$ (black) to that where $1 / \sigma_1 ( (A-zI )^{-1}) = 10^{-3}$ (red); i.e., to the boundary of the $10^{-3}$-pseudospectrum. Upper left is the Grcar matrix of size $n=50$, upper right is the transient_demo matrix of size $n=50$. Lower left is the sampling matrix of size $n=10$, lower right is a Jordan block of size $n=10$.

Theorems & Definitions (12)

• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof
• Corollary 5
• proof