$A \in M_n (\mathbb{R})$ and $A$ is invertible, let $S=\left\{(x_1,x_2,\cdots,x_n)^{'} \in R^{n}\mid\sum_{k=1}^{n}x_{k}^{2}=1\right\}$ and $T=\left\{Ax \mid x = (x_1,x_2,\cdots,x_n)^{'} \in S\right\}$, proof, there is an orthogonal matrix $Q$ and positive real numbers $\sigma_1,\cdots,\sigma_n$ such that $$ U=\left\{Qu \mid u = (u_1,u_2,\cdots,u_n)^{'} \in T\right\} = \left\{y= (y_1,y_2,\cdots,y_n)^{'} \in R^n\mid\sum_{k=1}^n\frac{y_k^2}{\sigma_k^2}=1\right\}. $$
My attempt,
$A$ is invertible so that $A^{\prime}A$ is positive-definite, there exists invertible matrix $T$ such that $T^{\prime}A^{\prime}AT = \mathrm{diag} \left \{ \lambda _1, \lambda _2, \cdots , \lambda _n \right \} $ where $\lambda _i >0$, and I have no idea what's next, in fact,I don't know how to start from the conclusion and solve the problem.
My question,
(1) the solution;
(2) Is this problem has any background infromation? I find that elements of $S$ is a "ball" of $R^n$ and the result $U$ also looks like a "ball".
Thanksfor your answer and any insights are welcome!
