Check if the function $h(A) = \lim\limits_{n \to +\infty} \frac{\text{tr}((A^A)^{n+1})}{\text{tr}((A^A)^{n})}$ is a valid matrix norm

Question

For a given function

$$h(A) = \lim\limits_{n \to +\infty} \dfrac{\text{tr}((A^A)^{n+1})}{\text{tr}((A^A)^{n})}$$

we have to check if it's a valid matrix norm.

I know that $A^A$ is defined as

$$A^A = \exp(A\ln(A))$$

where both $\exp(X)$ and $\ln(X)$ defined as infinite sum of series.

It's completely unclear though, how this definition may be of help to solve the problem. Not sure where to start and how to make any progress on this one. Maybe anyone could give an idea to the solution?

Answer 1

Let $\lambda_1, \ldots, \lambda_n$ be the eigenvalues of $A$ with multiplicities. The spectral mapping theorem implies that $\lambda_1^{\lambda_1}, \ldots, \lambda_n^{\lambda_n}$ are the eigenvalues of $A^A$, with the same multiplicities. Denote $x_i = \lambda_i^{\lambda_i}$.

WLOG assume that $|x_1| \le |x_2| \le \cdots \le |x_n|$.

\begin{align}\lim_{k\to\infty} \frac{\operatorname{Tr} (A^A)^{k+1}}{\operatorname{Tr} (A^A)^{k}} &= \lim_{k\to\infty} \frac{x_1^{k+1} + x_2^{k+1} \ +\cdots + x_n^{k+1}}{x_1^{k} + x_2^{k} \ +\cdots + x_n^{k}}\\ &= x_n \lim_{k\to\infty}\frac{\left(\frac{x_1}{x_n}\right)^{k+1} + \left(\frac{x_2}{x_n}\right)^{k+1} +\cdots + \left(\frac{x_{n-1}}{x_n}\right)^{k+1} + 1}{\left(\frac{x_1}{x_n}\right)^{k} + \left(\frac{x_2}{x_n}\right)^{k} +\cdots + \left(\frac{x_{n-1}}{x_n}\right)^{k} + 1}\\ &= x_n \end{align}

This is not a norm. For the identity matrix we have $h(I) = 1^1 = 1$ but $h(2I) = 2^2= 4$.