The modified bordering method to evaluate eigenvalues and eigenvectors of normal matrices (Q1911456)

An algorithm is presented for computing the characteristic polynomial of an \(n\times n\) normal matrix, using the elements of the last row and column and the eigenvalues and corresponding eigenvectors (but not explicitly the elements) of the top left \(n-1 \times n-1\) principal submatrix. The authors report that they have found this an effective method for computing eigenvalues and eigenvectors of a bordered matrix when sufficiently high precision is used, although it is known that well-conditioned matrix eigenvalues may be ill-conditioned zeros of the characteristic polynomial.