Schur Characteristic Polynomials

Schur

A characteristic polynomial $\Delta [z]$ is Schur if $\text{roots}(\Delta )\subset \mathbb{D}$.

Lemma

Let $\Delta [z]={\sum }_{i=0}^n{c}_i{z}^i$. If $\Delta$ is Schur, then $|c_n|>|c_0|$.

Note that this does not go the other way! We could have $|c_n|>|c_0|$ but the polynomial is not Schur.

Example 1:

$$z^3+z+3=\Delta [z]$$

where

• $c_n=c_3=1$
• $c_1=1$
• $c_0=3$

Then:

$$\begin{array}{r}|c_n|=|c_3|=1\\ |c_0|=3\end{array}⟹|c_n|<|c_0|⟹\Delta \text{ is NOT Schur!}$$

Example 2:

$$z^2+\frac{1}{2}z+\frac{1}{2}=\Delta [z]$$

Here, $|c_n|=1$ and $|c_0|=\frac{1}{2}$. Thus, $|c_n|>|c_0|$.

• However, if we factor out the characteristic polynomial, we can see that $\Delta z=(z+2)\left(z+\frac{1}{4}\right)$, so the polynomial is not Schur!
• This shows the limitations of the lemma – we still have to check our roots!

Checking if $\Delta [z]$ is Schur

Let’s consider something else. Let:

$$\Delta [z]=\sum _{i=0}^n{c}_i{z}^i,Q[z]=\sum _{i=0}^n{c}_{n-i}{z}^i$$

Example:

$$\begin{array}{r}\Delta [z]=2z^2+3z+4\\ Q[z]=4z^2+3z+2\end{array}$$

Goal: Find a lower-order polynomial for testing if $\Delta$ is Schur – we want to cancel out $c_0$ and then divide by $z$.