Espais vectorials de linealitzacions per a matrius polinomials

Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2023, Director: M. Eulàlia Montoro López[en] For eigenvalue problems of polynomial matrices we find that the classic solution method is linearization of the polynomial matrix. Reformulating the initial eigenvalue problem we obtain an expression for matrix pencils, that is, matrices of the form $\lambda X+Y, X, Y \in \mathbb{C}^{n \times n}$, which maintains the spectral structure. Within the framework of linear algebra and matrix theory, polynomial matrices are objects of recent study. In this work we focus on square regular polynomial matrices, i.e. with non-zero determinant. We introduce the basic concepts necessary to understand them, then we see what the linearization of regular polynomial matrices consist of and we define the "companion forms" or companion matrices. Finally, we study the vector spaces of linearizations; more specifically, how to construct two vector spaces of dense pencils in the set of linearizations