MATH319 Slides 15 Proof of reduction of order 17 Chapter 2: Solving linear systems by matrix theory

16 Constant coefficient case

If the $a_{j}$ are constant, or equivalently $A$ is constant, then the system is said to have constant coefficients, and the differential equation can be expressed as the feedback system

X=A\int X+\int U.

Definition (Companion matrix)

Let $a_{j}\in{\bf C}$ for $j=0,\dots,n-1$ . Then the $(n\times n)$ matrix

A_{c}=\begin{bmatrix}0&1&0&\dots\\ 0&0&1&\ddots\\ \vdots&\vdots&\ddots&\ddots\\ -a_{0}&-a_{1}&\dots&-a_{n-1}\end{bmatrix}

is called a companion matrix (as in 15; note the signs of the $a_{j}$ .)