TITLE: Singular values of a matrix written in controllable canonical form
QUESTION [9 upvotes]: Let the following equation represent a stable (marginally) dynamical system in discrete time domain
\begin{equation}
\mathbf{x}_{k+1} = \mathbf{A}\mathbf{x}_k + \mathbf{B}\mathbf{u}_k
\end{equation}
where $\mathbf{A}\in\mathbb{R}^{m\times m}$, $\mathbf{x}_k\in\mathbb{R}^m$, $\mathbf{B}\in\mathbb{R}^{m\times n}$ and $\mathbf{u}_k\in\mathbb{R}^{n}$. $\mathbf{A}$ is written in controllable canonical form. Since the system is stable (marginally), all the eigenvalues of $\mathbf{A}$ lie inside or on the unit circle. 

Given the system is stable, is it correct to say that at least $m-1$ singular values of $\mathbf{A}$ are less than or equal to unity and at most one singular value can be greater than $1$?

REPLY [4 votes]: A controller canonical form in control theory is just a companion matrix in linear algebra. It is known in linear algebra literature that for any $m\times m$ companion matrix (with $m\ge2$)
$$A=\pmatrix{
0 & 1 & 0 & \cdots & 0\\
0 & 0 & 1 & \cdots & 0\\
\vdots & \vdots & \vdots & \ddots & \vdots\\
0 & 0 & 0 & \cdots & 1\\
-a_0 & -a_1 & -a_2 & \cdots & -a_{m-1}},
$$
its middle $m-2$ singular values are given by $\sigma_2=\cdots=\sigma_{m-1}=1$ and its largest and smallest singular values are given by
$$
\sigma_1,\sigma_m=\sqrt{\frac{1+\|\mathbf a\|^2 \pm \sqrt{(1+\|\mathbf a\|^2)^2 - 4a_0^2\,}}2},
$$
where $\|\mathbf a\|$ is the Euclidean norm of $\mathbf a=(a_0,a_1,\ldots,a_{m-1})^T$. (See, e.g. Horn and Johnson, Matrix Analysis (2/e), p.197, 3.3.P13 or Alan Laub, Matrix Analysis for Scientists and Engineers, p.106, theorem 10.38.)
In your case, if the spectral radius of $A$ is at most $1$, we have $|a_0|\le1$. Therefore the only situation where $\sigma_1>1>\sigma_m$ fails to hold is that $|a_0|=1$ and $a_1=\cdots=a_{m-1}=0$.