Chapter 7, Problem 13E

Let
$A=\left(\begin{array}{ccc}4& 5& 2\\ 4& 5& 2\\ 0& 3& 6\\ 0& 3& 6\end{array}\right)$
Enter the matrix A in MATLAB and compute its singular values by setting $s=svd\left(A\right).$

1. How can the entries of s be used to determine the values ${\Vert A\Vert }_2$ and ${\Vert A\Vert }_F?$ Compute these norms by setting
   $p\text{ =}\text{norm}\left(\text{A}\right)$ and $q=\text{norm}\left(A,\text{'}fro\text{'}\right)$ and compare your results with s(l) and norm(s).
2. To obtain the full singular value decomposition of A, set
   $\left[U,D,V\right]=\text{svd}\left(A\right)$
   Compute the closet matrix of rank 1 to A by setting
   $B=s\left(1\right)*U\left(:,1\right)*V\left(:,1\right)\text{'}$
   How are the row vectors of B related to the two distinct row vectors of A?
3. The matrices A and B should have the same 2-norm. Why? Explain. Use MATLAB to compute ${\Vert B\Vert }_2$ and ${\Vert B\Vert }_F.$ In general, for a rank I matrix the 2-norm and the Frobenius norm should be equal. Why? Explain.