I want to know how to proof this theorem:
The cosets of $SO(n)$ in $GL^{+}(n,\mathbb{R})$ are the set of matrices
$$
A SO(n)=\{AQ|Q \in SO(n)\},A \in GL^{+}(n,\mathbb{R})
$$
It can be shown (using the polar form for matrices) that there is a bijection between the cosets of $SO(n)$ in $GL^{+}(n,\mathbb{R})$ and the set of $n \times n$ symmetric, positive, definite
matrices; these are the symmetric matrices whose eigenvalues are strictly positive.
Best Answer
If $A$ and $B$ lie inside the same coset, then $A=BQ$ for some $Q\in SO(n)$. Hence $AA^T=BB^T$ and this product is positive definite. Since each positive definite matrix has a unique positive definite square root, $P=(AA^T)^{1/2}$ is the unique positive definite representative of the coset $A(SO(n))$.