Suppose $A$ is an invertible real $n\times n$ matrix, and consider its (unique) polar decomposition $A=BJ$ where $B$ is positive definite symmetric and $J$ is orthogonal. Is it true that $AB=BA$?
Actually I am reading Audin, Damian's book Morse Theory and Floer Homology and the above statement is assumed in p.141. To be more precise, for a symplectic vector space $(E,\omega)$ with an inner product $(,)$, the matrix $A$ is determined by $(X,AY)=\omega(X,Y)$.
The claim is not true. For $$A=\begin{pmatrix}1 &1\\ 0 & 1\end{pmatrix}$$ we have $$B^2=AA^T=\begin{pmatrix}2 &1\\ 1 & 1\end{pmatrix}$$ and $AB^2\neq B^2A.$
The claim is true if the matrix $A$ is symmetric, or more generally normal, i.e. $AA^T=A^TA.$