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Regular and singular matrix polynomials P (λ) = k i=0 P i φ i (λ), P i ∈ R n×n given in an orthogonal basis φ 0 (λ), φ 1 (λ), . . . , φ k (λ) are considered. Following the ideas in [9], the vector spaces, called M 1 (P ), M 2 (P ) and DM(P ), of potential linearizations for P (λ) are analyzed. All pencils in M 1 (P ) are characterized concisely. Moreover, several easy to check criteria whether a pencil in M 1 (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 M 1 (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 L 1 , i.e. the results obtained in [9], and their generalized versions are pointed out.

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