Computing the Jordan Structure of an Eigenvalue

Mastronardi, Nicola;Van Dooren, Paul
(2017) SIAM Journal on Matrix Analysis and Applications — Vol. 38, n° 3, p. 949-966 (2017)

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Authors
  • Mastronardi, Nicola
    Author
  • Van Dooren, PaulUCLouvain
    Author
Abstract
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.
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Mastronardi, N., & Van Dooren, P. (2017). Computing the Jordan Structure of an Eigenvalue. SIAM Journal on Matrix Analysis and Applications, 38(3), 949-966. https://doi.org/10.1137/16M1083098 (Original work published 2017)