Computational Mathematics and Scientific Computing Seminar

The generalized null space decomposition and its implications for computing the Jordan form

Speaker: Michael Overton, NYU Courant

Location: Warren Weaver Hall 1302

Date: May 3, 2019, 10 a.m.


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.

Joint work with G.W. (Pete) Stewart and Nicola Guglielmi.