On the construction of nearest defective matrices to a normal matrix (Q1765931)

For an \(n\times n\) normal matrix \(A\) having \(n\) distinct eigenvalues a simple procedure for constructing a defective matrix nearest to~\(A\) with respect to the spectral norm is described. (An \(n\times n\) matrix is defective if it has fewer than \(n\) linearly independent eigenvectors.) This construction requires only a pair of eigenvalues of~\(A\) and their corresponding eigenvectors. By contrast, if \(A\) is not normal, the construction of its nearest defective matrix is a hard problem.