# A probelm of constructing an orthogonal matrix and $n$ positive real numbers

$$A \in M_n (\mathbb{R})$$ 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".

• You were almost sucessful with $A'A=P^2$ positive definite. Just observe that $P^{-1}A' AP^{-1}=I$ and therefore $AP^{-1}$ is an orthogonal matrix. Commented Dec 7, 2023 at 9:54