Here is a special $(N+1)\times N$ matrix:

$$A=\begin{pmatrix}a_1&a_2&a_3&\ldots&a_N\\b_1&0&0&\ldots&0\\0&b_2&0&\ldots&0\\0&0&b_3&\ldots&0\\\vdots&\vdots&\vdots&\ddots&0\\0&0&0&\ldots&b_N\end{pmatrix}_{(N+1)\times N}$$

Where the matrix elements $a_i$, $b_i$ are all real and positive.

Does it exist two orthogonal matrices $U_{(N+1)\times(N+1)}$ and $V_{N\times N}$, satisfying

$$UAV=\begin{pmatrix}0&0&0&\ldots&0\\\xi_1&0&0&\ldots&0\\0&\xi_2&0&\ldots&0\\0&0&\xi_3&\ldots&0\\\vdots&\vdots&\vdots&\ddots&0\\0&0&0&\ldots&\xi_N\end{pmatrix}_{(N+1)\times N}$$?

For the case $a_i\ll b_i$, I can find the answer (within the accuracy of $\frac{a_i}{b_i}$); but for the general case, I have no idea.

I am not sure if this problem is a kind of typical exercise in the linear algebra textbook.

Thanks for everyone who help me to solve this problem or give me some hints.

  • 1
    $\begingroup$ Just perform a singular value decomposition $A=U_1SV^T$ and permute the columns of $U_1$ cyclically to the right. Then take $U=U_1^T$. $\endgroup$ – user1551 Jul 6 '17 at 13:34
  • $\begingroup$ Thank you very much! That's exactly what I want! $\endgroup$ – Doggy Jul 7 '17 at 7:28

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