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 $$\lim_{m \to \infty} \left( \frac{A}{\lambda} \right)^m = \frac{v u^T}{u^T v}.$$
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 $\frac{vu^T}{u^T v}$ 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!