matrix with eigenvalues on the unit circle

Question

Suppose $A$ is an $n\times n$ matrix with complex entries such that there exists strictly positive constants $c_1<1<c_2$ so that $$c_1<\frac{\|A^Nx\|}{\|x\|}<c_2$$ for any integer $N\geq 1$ and $x\neq 0$

(a) Show that if $\lambda $ is an eigenvalue of $A$, then $|\lambda|=1$

(b) Show that $A$ is similar to a unitary matrix.

How would utilize the inequality? Let $\{v_1,\ldots,v_n\}$ be the eigenvectors of $A$ corresponding to eigenvalues $\lambda_1,\ldots,\lambda_n$ then $A^Nv=c_1\lambda_1^Nv_1+\ldots+c_n\lambda_n^Nv_n$. However, in here I already assumed $A$ is diagonalizable. Any hints/ideas? thanks

Answer 1

For part (a): Suppose $\lambda$ is an eigenvalue of $A$ and $v$ is a corresponding eigenvector.

Then, $A^Nv = \lambda^Nv$ for every integer $N \ge 1$. Hence, $\dfrac{\|A^Nv\|}{\|v\|} = \dfrac{\|\lambda^Nv\|}{\|v\|}= |\lambda|^N$.

Now, suppose $|\lambda| < 1$ or $|\lambda| > 1$. Can you show that $c_1 < \dfrac{\|A^Nx\|}{\|x\|} < c_2$ is violated for $x = v$ and some large integer $N$?

For part (b): Let $A = VJV^{-1}$ be the Jordan canonical form of $A$. From part (a), you already know that the diagonal entries of $J$ (the eigenvalues of $A$) have magnitude $1$. If $J$ is strictly diagonal, then $J^*J = JJ^* = I$ is trivial. So suppose $J$ has a Jordan block of size $2$ or larger and try to derive a contradiction. You can do this by constructing a vector $x \neq \vec{0}$ such that $\|A^Nx\|$ gets arbitrarily large as $N \to \infty$. Then, this vector $x$ will violate $c_1 < \dfrac{\|A^Nx\|}{\|x\|} < c_2$ for some large integer $N$.

For instance, in the specific case where $J = \begin{bmatrix}\lambda&1\\0&\lambda\end{bmatrix}$, set $x = Vy$, where $y = \begin{bmatrix}0 \\ 1\end{bmatrix}$. Then, we get $A^Nx = VJ^NV^{-1}Vy = VJ^Ny = V\begin{bmatrix}\lambda^N & N\lambda^{N-1} \\ 0& \lambda^N\end{bmatrix}\begin{bmatrix}0 \\ 1\end{bmatrix} = V\begin{bmatrix}N\lambda^{N-1}\\\lambda^N\end{bmatrix}$. Hence,

$\|A^Nx\|$ $= \left\|V\begin{bmatrix}N\lambda^{N-1}\\\lambda^N\end{bmatrix}\right\|$ $= \left\|\lambda^NV\begin{bmatrix}0\\1\end{bmatrix}+N\lambda^{N-1}V\begin{bmatrix}1\\0\end{bmatrix} \right\|$ $\ge N\left\|V\begin{bmatrix}1\\0\end{bmatrix}\right\| - \left\|V\begin{bmatrix}0\\1\end{bmatrix}\right\|$ $\to \infty$ as $N \to \infty$. (Note that $V\begin{bmatrix}1\\0\end{bmatrix} \neq 0$ since $V$ is invertible). Thus, $\|A^Nx\| \to \infty$ as $N \to \infty$.

I'll let you work out the general case where $J$ has several Jordan blocks with at least one that is of size $n_i \ge 2$.