Matrix analysis
Topics in matrix analysis
SIAM Journal on Matrix Analysis and Applications
About Hölder condition numbers and the stratification diagram for defective eigenvalues
Mathematics and Computers in Simulation - IMACS sponsored special issue: 1999 international symposium on computational sciences, to honor John R. Rice
A Chart of Backward Errors for Singly and Doubly Structured Eigenvalue Problems
SIAM Journal on Matrix Analysis and Applications
Linear Perturbation Theory for Structured Matrix Pencils Arising in Control Theory
SIAM Journal on Matrix Analysis and Applications
Eigenvalue patterned condition numbers: Toeplitz and Hankel cases
Journal of Computational and Applied Mathematics
Structured Eigenvalue Condition Numbers
SIAM Journal on Matrix Analysis and Applications
Structured Hölder Condition Numbers for Multiple Eigenvalues
SIAM Journal on Matrix Analysis and Applications
Eigenvalue condition numbers: Zero-structured versus traditional
Journal of Computational and Applied Mathematics
Structured Pseudospectra for Small Perturbations
SIAM Journal on Matrix Analysis and Applications
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Let $\lambda$ be a nonderogatory eigenvalue of $A\in\mathbb{C}^{n\times n}$ of algebraic multiplicity $m$. The sensitivity of $\lambda$ with respect to matrix perturbations of the form $A\leadsto A+\Delta$, $\Delta\in\boldsymbol{\Delta}$, is measured by the structured condition number $\kappa_{\boldsymbol{\Delta}}(A,\lambda)$. Here $\boldsymbol{\Delta}$ denotes the set of admissible perturbations. However, if $\boldsymbol{\Delta}$ is not a vector space over $\mathbb{C}$, then $\kappa_{\boldsymbol{\Delta}}(A,\lambda)$ provides only incomplete information about the mobility of $\lambda$ under small perturbations from $\boldsymbol{\Delta}$. The full information is then given by the set $K_{\boldsymbol{\Delta}}(x,y)=\{y^*\Delta x;$ $\Delta\in\boldsymbol{\Delta},$ $\|\Delta\|\leq1\}\subset\mathbb{C}$ that depends on $\boldsymbol{\Delta}$, a pair of normalized right and left eigenvectors $x,y$, and the norm $\|\cdot\|$ that measures the size of the perturbations. We always have $\kappa_{\boldsymbol{\Delta}}(A,\lambda)=\max\{|z|^{1/m};$ $z\in K_{\boldsymbol{\Delta}}(x,y)\}$. Furthermore, $K_{\boldsymbol{\Delta}}(x,y)$ determines the shape and growth of the $\boldsymbol{\Delta}$-structured pseudospectrum in a neighborhood of $\lambda$. In this paper we study the sets $K_{\boldsymbol{\Delta}}(x,y)$ and obtain methods for computing them. In doing so we obtain explicit formulae for structured eigenvalue condition numbers with respect to many important perturbation classes.