Find the singular decomposition of T i.e. orthonormal bases and scalars so $Tv=...$

Question

Find the singular decomposition of $$T=\begin{bmatrix} 0\ 0\ 4 \\ \frac{5}{2}\ \frac{-1}{2} \ 0 \\ \frac{-1}{2}\ \frac{5}{2} \ 0 \end{bmatrix}$$.

That is find an orthonormal basis $(e_1,e_2,e_3)$, another orthonormal basis $(f_1,f_2,f_3)$, and scalars $s_1,s_2,s_3$ such that $$Tv=s_1 \langle v,e_1\ \rangle f_1+s_2 \langle v,e_2\ \rangle f_2+s_3 \langle v,e_3\ \rangle f_3$$ for all $v \in \mathbb{R}^3$

Answer 1

Here's how I do it for a particular matrix: http://www.wolframalpha.com/input/?i=singular+value+decomposition+of+%7B%7B+0,+0,+4%7D,%7B5%2F2,+-1%2F2,+0%7D,+%7B-1%2F2,+5%2F2,+0%7D%7D