The Generalized Null Space Decomposition and its Relevance to the Computation of the Jordan Canonical Form

Michael Overton, New York University SCAIM Seminar
January 22, 2019 12:30 pm ESB 4133

The generalized null space decomposition (GNSD), better known as the staircase decomposition, is a unitary reduction of a general matrix A of order n to a block upper triangular form that reveals the structure of the Jordan blocks of A corresponding to a zero eigenvalue. The reduction was introduced by Kublanovskaya in 1968. It was extended first by Ruhe and then by Golub and Wilkinson, who based the reduction on the singular value decomposition. If A has large Jordan blocks, the complexity of these algorithms can approach the order of n^4. We outline an alternative algorithm, based on repeated updates of a QR decomposition of A, that is guaranteed to be of order n^3. Numerical experiments confirm the stability of this algorithm, which turns out to produce essentially the same form as that of Golub and Wilkinson. The effect of errors in A on the ability to recover the original structure is investigated empirically. Several applications are discussed, including the computation of the Drazin inverse.