Consider a square real non-negative matrix $\mathbf{A}$.
My question is: is there a way to write $\mathbf{A}^{t}\big({\mathbf{A}^{t}}^{T}\big)$ (where $^T$ is transpose and $^t$ is matrix t_th power) nicely from some decomposition of $\mathbf{A}$ or $\mathbf{A}\mathbf{A}^{T}$ or anything else.
I know I can SVD-decompose $\mathbf{A}=\mathbf{Q}\mathbf{D}\mathbf{V}^{T}$, from which I can get, for $t=1$ a SVD of $\mathbf{A}\mathbf{A}^{T}=\mathbf{Q}\mathbf{D}^{2}\mathbf{Q}^{T}$. But then for $t=2$ (and further) it does not seem to lead to anything.