Counting Fiedler pencils using diagrams – Federico Poloni (Dipartimento di Informatica, Universita’ di Pisa)


Fiedler pencils are a family of matrix pencils which generalizes the well-known companion matrix: they are *linearizations*, i.e., they provide a method to construct, given a matrix polynomial, a linear eigenvalue problem with the same eigenvalues and multiplicities. Fiedler pencils are constructed as products of special block matrices that act nontrivially only on two contiguous blocks. They have a rich structure that give rise to many combinatorial properties. In this talk, we introduce a notation that associates to each pencil a diagram that depicts its action on the blocks. Using this notation, we can obtain visual proofs of several statements in the theory, and we can solve several counting properties (such as “how many distinct Fiedler pencils with repetitions of a given dimension exist”). Among them, in particular, we are interested in counting Fiedler pencils associated to symmetric and palindromic matrix polynomials which preserve the same structure. This talk is based on a joint work with Gianna Del Corso. Il seminario si tiene in Sala Seminari Ovest, Dipartimento di Informatica

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