Start Date
April 2025
Location
3rd floor - Library
Abstract
We study block companion matrices C of size mn x mn where the m blocks, denoted Ak for k=1, .., m, are n x n matrices of a particular form. We explore specific forms of C for arbitrary m, n by looking at their corresponding directed graphs. Using a relabeling of the vertices, we produce a diagonal block form for C for some types of non-strongly connected Ak's. We found that, for some forms of Ak, the relabeling produces diagonal blocks of smaller matrices having the same form as C. These are then used to simplify the problem of finding the characteristic polynomial of C to that of finding characteristic polynomials of matrices having the same form as C but of smaller order.
Reducible Block Companion Matrices
3rd floor - Library
We study block companion matrices C of size mn x mn where the m blocks, denoted Ak for k=1, .., m, are n x n matrices of a particular form. We explore specific forms of C for arbitrary m, n by looking at their corresponding directed graphs. Using a relabeling of the vertices, we produce a diagonal block form for C for some types of non-strongly connected Ak's. We found that, for some forms of Ak, the relabeling produces diagonal blocks of smaller matrices having the same form as C. These are then used to simplify the problem of finding the characteristic polynomial of C to that of finding characteristic polynomials of matrices having the same form as C but of smaller order.
