 2

Throughout Question 2(a) and 2(b), write $U=[u_1,\dots ,u_n]$, $V=[v_1,\dots ,v_n]$, and $\Sigma =\mathrm{diag}({\sigma }_1,\dots ,{\sigma }_n)$. Then the supplied full singular value decomposition can also be written $A=U\Sigma V^T$, with $U^{T}U=V^{T}V=I_n$ and $[U,U_{\perp }]$ an $m\times m$ orthogonal matrix.