P. Petkov. The Jordan Canonical Form – Myths and Reality

Key Words: Jordan Canonical Form; algorithm of Kublanovskaya-Ruhe-Kågström; Weyr Canonical Form.

Abstract. The paper presents some comments about the development and implementation of numerical algorithms for finding the Jordan canonical form of a square matrix. A short history of the algorithm of Kublanovskaya-Ruhe-Kågström is given. This algorithm uses an orthogonal reduction to staircase form in order to find the Segre characteristic of the multiple eigenvalue (the dimensions of the Jordan blocks pertaining to this eigenvalue). It is noted that this algorithm actually finds the Weyr characteristic and Weyr canonical form of the original matrix. That is why the program of Kågström and Ruhe can determine reliably the numerical structure of the given matrix. It is argued that in most cases this program can produce an accurate result for the Jordan form although this is in contradiction with the opinion of many researchers working in the field of matrix computations. Three myths concerning the numerical determination of the Jordan form are discussed. An 8th order example is given which demonstrates that the program system MATLAB® (and LAPACK package) produce results for the multiple eigenvalues which contain large errors, while the Kågström-Ruhe algorithm finds these eigenvalues to full machine precision. In such cases the eigensystem problem is ill-conditioned and some regularization technic is necessary to use. It is insisted that the Kågström-Ruhe algorithm is well suited for such a purpose. The paper ends by the note of G. W. Stewart that“… as a mathematical probe the Jordan canonical form is still useful, and reports of its dead are greatly exaggerated”.