Starting from a quadratic in $z\in\mathbb{C}$ with real scalar coefficients $b,c$:
$z^2 + bz+c=0$
and using the Schur-Cohn recursion, I can get the following conditions on $a,b,c$ such that $|z|\leq 1$:
$|b|-1\leq c\leq 1$
What I would like to know if there is a simple matrix equivalent to these conditions..
So now with a matrix quadratic equation:
$z^2Iv + zBv + Cv=0$
where $B,C$ are non-commutable square matrices, $I$ is the identity matrix, and $v$ is a column vector. Is it possible to obtain some simple analogue matrix inequalities to ensure that $|z|\leq1$, in terms of the eigenvalues of $B$ and $C$?
I'm guessing it will be something of the form:
$|B|-I\leq C\leq I$