On vector spaces of linearizations for matrix polynomials in orthogonal bases
For researchers in numerical linear algebra and polynomial eigenvalue problems, this work generalizes a known framework to orthogonal bases, but the results are largely incremental extensions of existing theory.
The paper extends the theory of linearizations for matrix polynomials from monomial bases to orthogonal bases, characterizing all pencils in the vector space M1(P) and providing criteria for (strong) linearizations. It derives results on dimensions, genericity, and block-symmetric pencils using basic algebraic methods.
Matrix polynomials given in an orthogonal basis are considered. Following the ideas of Mackey et al. "Vector spaces of Linearizations for Matrix Polynomials" (2006), the vec- tor spaces, called M1(P), M2(P) and DM(P), of potential linearizations for P are analyzed. All pencils in M1(P) are characterized concisely. Moreover, several easy to check criteria whether a pencil in M1(P) is a (strong) linearization of P are given. The equivalence of some of them to the Z-rank-condition (see Mackey et al. 2006) is pointed out. Results on the vector space dimensions, the genericity of linearizations in and the form of block-symmetric pencils are derived in a new way on a basic algebraic level. Throughout the paper, structural resemblances between the matrix pencils in L1 , i.e. the results obtained in Mackey et al. 2006, and their generalized versions are pointed out.