An eigenvector is defined by $$ \lambda \mathbf{v} = A\mathbf{v}.\tag{1} $$ But suppose I change this to $$ \lambda \mathbf{v} = A\mathbf{v}^\alpha,\tag{2} $$ for real $\alpha\ne 1$, where $\mathbf{v}^\alpha$ means a vector with elements $v_i^\alpha$, along with the normalisation condition $$ \sum_i v_i = 1.\tag{3} $$ (Specifying the normalisation condition is necessary, because equation 2 is nonlinear.)
Let us call vectors that satisfy equations 2 and 3 "stretched eigenvectors", unless they have another name. I am wondering if they are studied, and if so, what their properties are. They seem somehow related to the q-exponential and q-logarithm in Tsallis statistics (with $q=\alpha$), and if that's the case it would be particularly nice to know if that connection has been worked out.
I am mostly interested in the Perron-Frobenius case, where $A$ has real, non-negative entries. Then when $\alpha=1$ there exists an eigenvector $\mathbf{v}$ with non-negative entries and the corresponding eigenvalue $\lambda$ is also real and non-negative. If the entries of $A$ are positive then this is unique.
I am interested in whether any aspects of Perron-Frobenius and its related theorems can be generalised to cases where $\alpha\ne 0$. It seems for example that the uniqueness property can go away when $\alpha>1$, but perhaps there is some sense in which it can be recovered by changing the conditions.
My motivation for this is the study of sub-exponential and super-exponential replicators, which leads to a dynamical system whose fixed points are given by equations 2 and 3.
