#### Title

A Polynomial in A of the Diagonalizable and Nilpotent Parts of A

#### Date

6-1-2017 10:30 AM

#### End Time

1-6-2017 10:45 AM

#### Location

RWEC 107

#### Department

Mathematics

#### Session Chair

Matthew Nabity

#### Session Title

Mathematics Senior Project presentations

#### Presentation Type

Presentation

#### Faculty Sponsor(s)

Scot Beaver

#### Abstract

Any square matrix $A$ can be decomposed into a sum of the diagonal ($D_A$) and nilpotent ($N_A$) parts as $A=D_A+N_A$. The components $D_A$ and $N_A$ commute with each other and with $A$. For many matrices $A,\,B$, if $B$ commutes with $A$, then $B$ is a polynomial in $A$; this holds for $D_A$ and $N_A$. Following a Herbert A. Medina preprint, this paper shows how to construct the polynomials $p(A)=N_A$ and $q(A)=D_A$. Further, the Jordan canonical form $J$ is a conjugate $QAQ^{-1}$ of $A$; this paper demonstrates that the conjugation relating $J$ and $A$ also relates $N_A$ and $N_J$ and $D_A$ and $D_J$, respectively.

#### Recommended Citation

Wilson, Kathryn, "A Polynomial in A of the Diagonalizable and Nilpotent Parts of A" (2017). *Academic Excellence Showcase Schedule*. 227.

https://digitalcommons.wou.edu/aes_event/2017/all/227

A Polynomial in A of the Diagonalizable and Nilpotent Parts of A

RWEC 107

Any square matrix $A$ can be decomposed into a sum of the diagonal ($D_A$) and nilpotent ($N_A$) parts as $A=D_A+N_A$. The components $D_A$ and $N_A$ commute with each other and with $A$. For many matrices $A,\,B$, if $B$ commutes with $A$, then $B$ is a polynomial in $A$; this holds for $D_A$ and $N_A$. Following a Herbert A. Medina preprint, this paper shows how to construct the polynomials $p(A)=N_A$ and $q(A)=D_A$. Further, the Jordan canonical form $J$ is a conjugate $QAQ^{-1}$ of $A$; this paper demonstrates that the conjugation relating $J$ and $A$ also relates $N_A$ and $N_J$ and $D_A$ and $D_J$, respectively.