A new similarity transformation method for eigenvalues and eigenvectors

Abstract

A new method of finding the eigenvalues and eigenvectors of an arbitrary complex matrix is presented. The new method is a similarity transformation method which transforms an arbitrary N × N matrix to a Jordan canonical form in N-1 or less transformations. Each transformation matrix is a matrix function-the matrix sign function with a ± added to the main diagonal elements. Using this matrix function as a similarity transformation gives a block diagonal form which is a reduced form of the transformed matrix. As the Jordan canonical form is found, the eigenvectors are simultaneously found since the product of transformation matrices must be a matrix of eigenvectors. The theoretical development of the new method and a computational scheme with examples are given. In the examples, the computational scheme is applied successfully to matrices which have characteristics that cause problems for most numerical techniques.