## Structure preserving deflation of infinite eigenvalues in structured pencils

Volker Mehrmann and Hongguo Xu

### Abstract

The long standing problem is discussed of how to deflate the part associated with the eigenvalue infinity in a structured matrix pencil using structure preserving unitary transformations. We derive such a deflation procedure and apply this new technique to symmetric, Hermitian or alternating pencils and in a modified form to (anti)-palindromic pencils. We present a detailed error and perturbation analysis of this and other deflation procedures and demonstrate the properties of the new algorithm with several numerical examples.

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### Key words

structured staircase form, structured Kronecker canonical form, symmetric pencil, Hermitian pencil, alternating pencil, palindromic pencil, linear quadratic control, $H_\infty$ control

### AMS subject classifications

65F15, 15A21, 93B40

### Links to the cited ETNA articles

 [5] Vol. 26 (2007), pp. 1-33 Ralph Byers, Volker Mehrmann, and Hongguo Xu: A structured staircase algorithm for skew-symmetric/symmetric pencils

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