A formula for the 2-norm distance from a matrix to the set of matrices with multiple eigenvalues
The author proves that the 2-norm distance from an \(n\times n\) matrix \(A\) to the matrices that have a multiple eigenvalue \(\lambda\) can be computed as the maximum with respect to \(\gamma\) of the second-lowest singular value of the \(2n\times 2n\) matrices with block-diagonal \(A-\lambda I\) in both entries and off-diagonal elements \(0\) and \(\gamma I\). Therefore, he can identify the 2-norm distance from \(A\) to the set of matrices with multiple eigenvalues -- and he is also able to give a constructive description of a minimizing matrix.
scientific article; zbMATH DE number 1355442