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.