I can't figure out if this is true:
Suppose $A^T=-A$ and that the symmetric matrix $AA^T$ is a positive definite (so diagonalizable, lets say $AA^T=O\Lambda O^T$, with all eigenvalues positive). Therefore, we can define a square root by
$$\sqrt{AA^T}=O\sqrt{\Lambda}O^T,$$
where $\sqrt{\Lambda}=\textrm{diag}(\sqrt{\lambda_1},\ldots\sqrt{\lambda_n})$, $\lambda_i$ eigenvalues of $AA^T$. One can see that $A$ commutes with $AA^T$, once $A^T=-A$. But I can't see if it is true that:
$$A\sqrt{AA^T}=\sqrt{AA^T}A$$
In other words, does $A$ commute with $AA^T$ implies $A$ commutes with $\sqrt{AA^T}$?
Any help will be appreciated.
Best Answer
In more generality, if $B$ is positive semidefinite and $AB=BA$, then $A\sqrt B=\sqrt B\,A$. The key observation is that there exists a polynomial $p\in\mathbb{R}[x]$ such that $\sqrt B=p(B)$. Then we have $$ AB^2=(AB)B=(BA)B=B(AB)=B(BA)=B^2A; $$ similarly we deduce that $AB^n=B^nA$ for any $n\in\mathbb{N}$, and so $Ap(B)=p(B)A$ for any polynomial.
The existence of the required polynomial is shown as follows: as $B$ is positive semidefinite, it is diagonalizable, so $B=SDS^{-1}$ with $D$ diagonal. Now choose a polynomial $p$ such that $p(d_{jj})=\sqrt{d_{jj}}$. Then $$ \sqrt{B}=S\sqrt{D}S^{-1}=Sp(D)S^{-1}=p(SDS^{-1})=p(B). $$