Reducible Block Companion Matrices
Start Date
1-8-2024 12:00 PM
End Date
1-8-2024 12:15 PM
Location
ALT 208
Abstract
We study block companion matrices Cm,n where the blocks Ak, k=1, ..., m, are prescribed n x n matrices of a particular form. In this project, we explore specific forms of Cm,n for arbitrary m,n ≥ by looking at their corresponding directed graphs. In particular, we provide a necessary and sufficient condition for the strong connectivity of the digraph of Cm,n in terms of the strong connectivity of the digraphs of the blocks Ak. Using a relabeling of the vertices, we produce a diagonal block form for Cm,n for certain non-strongly connected Ak whose strong components are disjoint. We found that, for some forms of Ak, the relabeling produces diagonal blocks of smaller matrices Cm̂,n̂ where m̂ ≤ m and n̂ ≤ n. These are then used to simplify the problem of finding the characteristic polynomial of Cm,n to that of finding characteristic polynomials of smaller order Cm̂,n̂.
Reducible Block Companion Matrices
ALT 208
We study block companion matrices Cm,n where the blocks Ak, k=1, ..., m, are prescribed n x n matrices of a particular form. In this project, we explore specific forms of Cm,n for arbitrary m,n ≥ by looking at their corresponding directed graphs. In particular, we provide a necessary and sufficient condition for the strong connectivity of the digraph of Cm,n in terms of the strong connectivity of the digraphs of the blocks Ak. Using a relabeling of the vertices, we produce a diagonal block form for Cm,n for certain non-strongly connected Ak whose strong components are disjoint. We found that, for some forms of Ak, the relabeling produces diagonal blocks of smaller matrices Cm̂,n̂ where m̂ ≤ m and n̂ ≤ n. These are then used to simplify the problem of finding the characteristic polynomial of Cm,n to that of finding characteristic polynomials of smaller order Cm̂,n̂.
