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Eigenvalues & Eigenvectorshard
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Let λ\lambdaλ be a simple eigenvalue of a matrix AAA with right eigenvector xxx (normalized to ∥x∥2=1\|x\|_2=1∥x∥2​=1) and left eigenvector yyy (normalized to ∥y∥2=1\|y\|_2=1∥y∥2​=1). The condition number s(λ)−1s(\lambda)^{-1}s(λ)−1 of the eigenvalue λ\lambdaλ measures its sensitivity to perturbations. What is s(λ)s(\lambda)s(λ)?