Let $A$ be a positive square matrix. Perron-Frobenius theory says that there exist $\lambda,v$ with $Av=\lambda v$ and $\lambda$ equals the spectral radius of $A$, $\lambda$ is simple, and $v$ is positive.

Now consider also the *left* Perron eigenvector $u^T A=\lambda u^T$. Another result of Perron-Frobenius theory is that

$$\lim_{m\to \infty} \frac{A^m}{\lambda^m} = \frac{v u^T}{u^T v}.$$

Suppose $\|v\|=1$. The above result says that the "correct" normalization for u is $u^T v=1$ rather than the more usual $u^T u=\|u\|^2=1$. This motivates the question: **what is the significance of the ratio**

$$\frac{u^T v}{u^T u} ?$$

Are there matrices $A$ for which this ratio is arbitrarily large? Arbitrarily small? Does this ratio determine any properties of $A$? Note that if $A$ is symmetric, then $u=v$ and this ratio is always equal to $1$, but that's not the case in general for arbitrary $A$. Could it be the case that this ratio is measuring how far $A$ is from being symmetric?

Note too that this normalization is necessary so that the limit $\frac{v u^T}{u^T v}$ is a projection matrix (i.e. that its only non-zero eigenvalue is one). In this context, I understand why the normalization is necessary, but I'm interested in *the amount of normalization necessary with respect to the length of $u$*.

Any pointers appreciated. Thanks!

**EDIT** In the comments, it is argued that the real quantity of interest in this setup is

$$\frac{\left( u^T v \right)^2}{\left(u^T u \right) \left( v^T v \right)}.$$

This quantity is also of interest to me, and an acceptable replacement for my original question.