# IP 12

IP 12. **Reduction of quadratic matrix polynomials to triangular form **

*Chair: Froilán M. Dopico*

There is no analogue of the generalized Schur decomposition for quadratic matrix polynomials in the sense that Q(λ)= λ^{2}A_{2} + λA_{1} + A_{0} ∈ C[λ]^{n×n} cannot, in general, be reduced to triangular form T (λ)= λ[{2]I + λT_{1} + T_{0} by equivalence transformations independent of λ. We show that there exist matrix polynomials E(λ), F (λ) with constant determinants such that E(λ)Q(λ)F (λ)= T (λ) is a triangular quadratic matrix polynomial having the same eigenstructure as Q(λ).

For any linearization λI_{2n} − A of Q(λ) with A_{2} nonsingular, we show that there is a matrix U ∈ C^{2n×n} with orthonormal columns such that S =[U AU] is

nonsingular and S^{-1}( λI_{2n} −A)S is a companion linearization of an n × n triangular quadratic T (λ)= λ^{2}I + λT_{1} + T_{0} equivalent to Q(λ). Finally, we describe a numerical procedure to compute U and T (λ).

Françoise Tisseur

School of Mathematics, The University of Manchester, UK