Philip Saltenberger

Heike Faßbender, Philip Saltenberger

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.

Heike Fassbender, Philip Saltenberger

In this paper, we introduce a new family of equations for matrix pencils that may be utilized for the construction of strong linearizations for any square or rectangluar matrix polynomial. We provide a comprehensive characterization of the resulting vector spaces and show that almost every matrix pencil therein is a strong lineariza- tion regardless whether the matrix polynomial under consideration is regular or sin- gular. These novel "ansatz spaces" cover all block-Kronecker pencils as a subset and therefore contain all Fiedler pencils modulo permutations. The important case of square matrix polynomials is examined in greater depth. We prove that the intersection of any number of block-Kronecker ansatz spaces is never empty and construct large subspaces of blocksymmetric matrix pencils among which still almost every pencil is a strong linearization. Moreover, we show that the original ansatz spaces L1 and L2 may essentially be recovered from block-Kronecker ansatz spaces via pre- and postmultiplication, respectively, of certain constant matrices.