A and B have same singular values

[Demon117]

Homework Statement

Suppose A is an m x n matrix and B is the n x m matrix obtained by rotating A ninety degrees clockwise on paper (not exactly a standard mathematical transformation!). Do A and B have the same singular values? Prove the answer is yes or give a counterexample.

The Attempt at a Solution

I have not been able to come up with a counterexample, so I am assuming the answer is yes. But I do not know how to prove that. What properties of Singular Values are there that deal with operations on matrices, like transposing and row-swapping, etc.?

EDIT: Example-

A = $$\left(\begin{array}{cc} -1 & 2 \\3 & 1 \\ 2 & -1 \end{array}\right)$$
B = $$\left(\begin{array}{ccc} 2 & 3 & -1 \\-1 & 1 & 2 \end{array}\right)$$

It follows that $$A^{T}A$$ = $$BB^{T}$$, so naturally the eigenvalues would be the same for these. Furthermore, the singular values of A and B would have to be the same as well. This is a good idea of what I am trying to prove but I cannot think of how this would be proved in the general case for m x n and n x m matrices A & B...

 

[vela]

The second column of B is upside down in your example.

I don't know if this helps, but you can say

$$B = \left[\begin{pmatrix}-1 & 2 \\3 & 1 \\ 2 & -1 \end{pmatrix} \begin{pmatrix}0 & 1 \\1 & 0 \end{pmatrix}\right]^T$$

 

[Demon117]

The second column of B is upside down in your example.

I don't know if this helps, but you can say

$$B = \left[\begin{pmatrix}-1 & 2 \\3 & 1 \\ 2 & -1 \end{pmatrix} \begin{pmatrix}0 & 1 \\1 & 0 \end{pmatrix}\right]^T$$

Umm, I don't buy into that. . . According to the definition of B it seems like the second column of B is in fact correct in my example. . . I am not seeing what you are referring to.

 

[vela]

Oh, sorry! I rotated counter-clockwise. How about

$$B = A^T\begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix}$$?

Again, I just want to point out, I don't know how the proof will work out, but this is a way you might be able to generalize for the mxn case what the transformation is.

 

[wisvuze]

I'm pretty sure A and B have the same singular values if and only if you can find isometries P and Q such that A = PBQ. The transposing operation and the row swapping operation are both isometries