Computing the Jordan Structure of an Eigenvalue

In this paperwe revisit the problem of finding an orthogonal similarity transformation that puts an n×n matrix A in a block upper-triangular form that reveals its Jordan structure at a particular eigenvalue λ0. The obtained form in fact reveals the dimensions of the null spaces of (A−λ0I)i at that eigenvalue via the sizes of the leading diagonal blocks, and from this the Jordan structure at λ0 is then easily recovered. The method starts from a Hessenberg form that already reveals several properties of the Jordan structure of A. It then updates the Hessenberg form in an efficient way to transform it to a block-triangular form in O(mn2) floating point operations, where m is the total multiplicity of the eigenvalue. The method only uses orthogonal transformations and is backward stable. We illustrate the method with a number of numerical examples.