The numerical Jordan form

In this expository paper, we discuss the properties and implementation of the numerical Jordan and Weyr canonical forms as computed by the algorithm of Kublanovskaya-Ruhe-Kågström. In contrast to the widespread negative opinion toward the numerical computation and implementation of the Jordan form, we emphasize its usefulness in the solution of some important problems of matrix analysis. The numerical Jordan form is defined by regularization of an ill-posed eigenvalue problem and, as opposed to the theoretical case, is not sensitive to small changes of the matrix elements. This makes possible its stable computation in the case of well-conditioned numerical structure. A short description of the algorithm of Kågström and Ruhe is presented and some examples are given to illustrate its performance. Also, an alternative numerical algorithm for finding the Jordan form proposed recently by Zeng and Li is briefly commented. Several misconceptions about the Jordan form are discussed and some ways to improve the existing numerical algorithms are briefly considered. It is asserted that reliable estimation of eigenvalue accuracy and sensitivity is impossible without knowing the Jordan structure of the matrix. As a case study, we present the computation of defective multiple eigenvalue sensitivity when the use of Jordan form is indispensable.