An operator $T ∈ L(V )$ is said to preserve angles if $T$ is invertible, and for all nonzero $u, v ∈ V$ we have $${(T(u), T(v)) \over {∥T(u)∥∥T(v) ∥}} = {(u, v) \over {∥u∥ ∥v∥}} $$
(a) Prove that if operators $T_1, T_2$ preserve angles, then $T_1T_2$ and $T_1^{−1}$ also preserve angles.
To tackle this problem I am thinking to show following facts:
Suppose $ V\in \mathbb R^n $ then $T_1 \in L(V)$ and $T_2 \in L(V)$ and $T_1T_2: \mathbb R^n \rightarrow \mathbb R^n$.
Assume there exists the positive number $\lambda > 0 $ and
(i) $T_1T_2(v) · T_1T_2(w) = λ^2(v · w)$ for all $v, w ∈ \mathbb R^n$
(ii) $ ||T_1T_2(v)|| = λ||v||$ for all $v ∈ \mathbb R^n$
Then I show $${(T_1T_2(u), T_1T_2(v)) \over {∥T_1T_2(u)∥∥T_1T_2(v) ∥}} = {(u, v) \over {∥u∥ ∥v∥}} $$
Please correct me if I am wrong. Hints and helps are greatly appreciated.
Same approach applies to $T^{-1}$.
(b) Suppose $T$ preserves angles. Let $T = SP$ be the polar decomposition, where $S$ is an isometry, and $P$ is a positive operator. Prove that $P$ preserves angles.
I don't have any ideas for this one. Please direct me how to approach this problem.
Thanks!
Hint: For the polar decomposition problem, note that since $S$ is an isometry, it preserves angles. Then, use both results from (a).