## A structured staircase algorithm for skew-symmetric/symmetric pencils

Ralph Byers, Volker Mehrmann, and Hongguo Xu

### Abstract

We present structure preserving algorithms for the numerical computation of structured staircase forms of skew-symmetric/symmetric matrix pencils along with the Kronecker indices of the associated skew-symmetric/symmetric Kronecker-like canonical form. These methods allow deflation of the singular structure and deflation of infinite eigenvalues with index greater than one. Two algorithms are proposed: one for general skew-symmetric/symmetric pencils and one for pencils in which the skew-symmetric matrix is a direct sum of $0$ and $\mathcal{J}=\left[\begin{array}{cc}0 & I \\ -I & 0 \end{array}\right]$. We show how to use the structured staircase form to solve boundary value problems arising in control applications and present numerical examples.

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### Key words

structured staircase form, linear-quadratic control, $H_\infty$ control, structured Kronecker canonical form, skew-symmetric/symmetric pencil, skew-Hamiltonian/Hamiltonian pencil

### AMS subject classifications

65F15, 15A21, 93B40

### Links to the cited ETNA articles

 [35] Vol. 13 (2002), pp. 106-118 Volker Mehrmann and David Watkins: Polynomial eigenvalue problems with Hamiltonian structure