Let $A$ be an entrywise nonnegative matrix and for any $r>0$, $A^{\circ r} = [a_{ij}^{r}]$. My question is what properties of matrix $A$ are preserved for all $r>0$ or for all $r$ in some interval ? For example : if I want to show the matrix $A^{\circ r}$ is nonsingular for all or some $r$, how can I proceed such type of questions ? Are there any integral, summation, product or limits formulae for the function $x \rightarrow x^{r}$ for all $x \in (0,\infty)$ and a fixed $r>0$, which can be applicable for such kind of questions. Same questions are for the matrix $B = \int_{a}^{b} a_{ij}(t)$, where $A(t) = [a_{ij}(t)]$ is a given matrix for any $t \in \mathbb{R}$.
Any important relative links for books or papers or examples will be appreciated. Thanks in advance.
Let $A$ be a $n\times n$ entrywise nonnegative matrix that is also PSD and let $\alpha$ be a real number.
If $\alpha\geq n-2$, then $A^{o\alpha}$ is PSD; moreover $n-2$ is the best bound. cf.
Fitzgerald, Horn: "On fractional Hadamard powers of PSD matrices".