Computing the Jordan structure of an eigenvalue

In this paper we revisit the problem of finding an orthogonal similarity transformation that puts an $n\times n$ matrix $A$ in a block upper-triangular form that reveals its Jordan structure at a particular eigenvalue $\lambda_0$. The obtained form in fact reveals the dimensions of the null spaces of $(A-\lambda_0 I)^i$ at that eigenvalue via the sizes of the leading diagonal blocks, and from this the Jordan structure at $\lambda_0$ 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 ${\cal O}(mn^2)$ 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.