Let $E,A$ be $m\times n$ matrices. Then I need to show there exists $U,V$ orthogonal of suitable order such that $$UEV=\begin{bmatrix}E_0&E_{k-1}&\times&\times&\times\\ &0&E_{k-2}&\times&\times\\&& \ddots&\ddots&\vdots\\ &&&0&E_1\\&&&&0 \end{bmatrix}$$
$$UAV=\begin{bmatrix}A_0&\times&\times&\times&\times\\&A_{k-1}&\times&\times&\times\\&& \ddots&\ddots&\vdots\\ &&&A_2&\times\\&&&&A_1\end{bmatrix}$$
where $\times$ stands for matrix of no-interest, $E_0$ is of full row rank, $A_i$ is of full column rank for $i=1,\dots,k-1;k\le n$
Here is what I did:
First choose $U_1,V_1$ orthogonal to make a row and column compression for $E$ and $A$ respectively, we get $U_1E=\begin{bmatrix}\hat{E}\\0\end{bmatrix}$ and $AV_1=\begin{bmatrix}A_1&0\end{bmatrix}$ where $\hat{E}$ is full row rank and $A_1$ is full column rank. I am not getting why $E$ and $A$ has different form after decomposition and why so many steps are needed if someone gets full row and column rank of decomposed $E$ and $A$ in first step? thanks for helping.
Actually I want a decomposition of the fixed matrix $A,E$ with variable $\lambda$: $\lambda E-A$, my question is if I apply first row compression of $E$ and then column compression of the remaining matrix $A$ will I get above form? Thanks for helping.