Matrices are the most common representations of graphs. They are also used for the representation of algebras and cluster algebras. This paper shows some properties of matrices in order to facilitate the understanding and locating symmetrizable matrices with specific characteristics, called positive quasi-Cartan companion matrices. Here, symmetrizable matrix are those which are symmetric when m...

Various generalizations of companion matrices to companion pencils are presented. Companion matrices link to monic polynomials, whereas companion pencils do not require monicity of the corresponding polynomial. In the classical companion pencil case (A,B) only the coefficient of the highest degree appears in B’s lower right corner. We will show, however, that all coefficients of the polynomial ...

In this talk, we give a criterion for a companion matrix to be reducible and give a complete characterization of reducible companion matrices. We also consider properties of the numerical ranges of reducible companion matrices. To be more precise, if C is an n-by-n reducible companion matrix which is unitarily equivalent to the direct sum of A and B, we study properties of the numerical ranges ...

A common method for computing the zeros of an n-th degree polynomial is to compute the eigenvalues of its companion matrix C. A method for factoring the shifted companion matrix C − ρI is presented. The factorization is a product of 2n − 1 essentially 2 × 2 matrices and requires O(n) storage. Once the factorization is computed it immediately yields factorizations of both (C − ρI) and (C − ρI). ...