Let $M$ be a non sparse matrix of real non negative values. Let $ A = M M^T $.
What can I say about the powers of $A$? Are they also symetric? Is possible to calculate the powers of $A$ only with $M$?
Since $A$ is always symetric. Decomposing $A$ with Cholesky ($A = L L^T$, being $L$ a lower diagonal matrix) gives me any improvement to calculate the powers of $A$?
Is possible to find the Cholesky decomposition of $A$ (that is $L$ lower triangular matrix) only with $M$?
The powers of a symmetric matrix are always symmetric: $(A^n)^T = (AAA...A)^T=A^TA^TA^T...A^T=AAA...A=A^n$ (here I have used the property $(BC)^T=C^TB^T$ and its generalization to an arbitrary number of factors). The most effective way to calculate the $n$-th power of a matrix is to use the spectral theorem: $A=Q^TDQ$, where $D$ is diagonal and $Q$ is orthogonal; it is easy to see that $A^n = Q^TD^nQ$. (for example in the case $n=2$ we have: $A^2=AA=Q^TDQQ^TDQ=Q^TD(QQ^T)DQ=Q^TDIDQ=Q^TDDQ=Q^TD^2Q$, where $I$ is the identity matrix).