Quote by Stephen Tashi
Are you saying that vector 'a' will be chosen so that the vector Ua will be 1 at the jth component iff the largest singular value occurs in S at location S[j][j] and the vector Ua will be zero elsewhere?
In these two examples, do we have the same matrix for A'D but different answers for the maximum angle? (My 4D intuition isn't good, so I'm not sure.)
Example 1: [itex] A = \begin{pmatrix} \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{3}}\\ \frac{2}{\sqrt{15}}\\ \frac{1}{\sqrt{15}} \end{pmatrix} ,\ D = \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ 0 & 0 \\ 0 & 0 \end{pmatrix} [/itex]
Example 2: [itex] A = \begin{pmatrix} \frac{1}{\sqrt{3}} \\ \frac{1}{\sqrt{3}}\\ \frac{1}{\sqrt{6}}\\ \frac{1}{\sqrt{6}} \end{pmatrix} ,\ D = \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ 0 & 0 \\ 0 & 0 \end{pmatrix} [/itex]

As I solved this example, since A'*D is the same, so are a and b, actually a=1 and b=[1/sqrt(2),1/sqrt(2)]. But x1=A1≠x2=A2. However the angle is the same, since <x,y>=(Ua)'*S*(Vb) is totally determined by A'*D. Seemingly logical.