On vector spaces of linearizations for matrix polynomials in orthogonal bases

Regular and singular matrix polynomials P(λ)=∑i=0kPiϕi(λ),Pi∈Rn×n given in an orthogonal basis ϕ0(λ),ϕ1(λ),…,ϕk(λ) are considered. Following the ideas in [9], the vector spaces, called M1(P), M2(P) and DM(P), of potential linearizations for P(λ) 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(λ) are given. The equivalence of some of them to the Z-rank-condition [9] is pointed out. Results on the vector space dimensions, the genericity of linearizations in M1(P) and the form of block-symmetric pencils are derived in a new way on a basic algebraic level. Moreover, an extension of these results to degree-graded bases is presented. Throughout the paper, structural resemblances between the matrix pencils in L1, i.e. the results obtained in [9], and their generalized versions are pointed out.