Reducible Block Companion Matrices
Start Date
August 2024
End Date
August 2024
Location
ALT 208
Abstract
We study block companion matrices where the blocks , , are prescribed matrices of a particular form. In this project, we explore specific forms of for arbitrary by looking at their corresponding directed graphs. In particular, we provide a necessary and sufficient condition for the strong connectivity of the digraph of in terms of the strong connectivity of the digraphs of the blocks . Using a relabeling of the vertices, we produce a diagonal block form for for certain non-strongly connected whose strong components are disjoint. We found that, for some forms of , the relabeling produces diagonal blocks of smaller matrices where and . These are then used to simplify the problem of finding the characteristic polynomial of to that of finding characteristic polynomials of smaller order .
Reducible Block Companion Matrices
ALT 208
We study block companion matrices where the blocks , , are prescribed matrices of a particular form. In this project, we explore specific forms of for arbitrary by looking at their corresponding directed graphs. In particular, we provide a necessary and sufficient condition for the strong connectivity of the digraph of in terms of the strong connectivity of the digraphs of the blocks . Using a relabeling of the vertices, we produce a diagonal block form for for certain non-strongly connected whose strong components are disjoint. We found that, for some forms of , the relabeling produces diagonal blocks of smaller matrices where and . These are then used to simplify the problem of finding the characteristic polynomial of to that of finding characteristic polynomials of smaller order .
