A new similarity transformation method for eigenvalues and eigenvectors



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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 x N matrix to a Jordan canonical form in N-l or less transformations. Each transformation matrix is a matrix function— the matrix sign function with a [plus-minus]1 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.