An upper bound for the spectral condition number of a diagonalizable matrix
A diagonalizable \(n\times n\) matrix \(A\), a transformation matrix \(P\) (defined by \(P^{-1}AP= J\), where \(J=\text{diag}(\lambda_1,\lambda_2,\dots,\lambda_n))\), and the spectral condition number \(\kappa(A)= \min_G |P|_2|P^{-1}|_2\), \(G=\{P\mid P^{-1}AP=J\}\), are considered. An upper bound of \(\kappa(A)\) is obtained from the elements of \(T\), the exact Schur form of \(A\). \(\kappa(A)\) is used to find the error bounds of eigenvalues and thus to estimate the accuracy of the computed eigenvalues. A numerical example is presented for illustration.
scientific article; zbMATH DE number 1040639