I'm currently facing a problem where I have $N$ matrices $ \{A_1, A_2, \dots, A_N \} $ and I want to find a way to calculate matrices $H$ and $E$ such that
$ \exists H \: \exists E\: \forall i \in [1, N] \exists \Delta_i \in \mathcal{W}:A_i = H \Delta_1 E $
where $ \mathcal{W} = \{\delta \mid \delta \in \mathbb{R}^{p \times q}, || \delta||_p \le 1 \}$
Therefore I want to find a left and a right common factor to all matrices, such that the residual matrix $\Delta_i$ $p$-norm is bounded.
An obvious necessary condition is that the sizes of all the $A_i$ is the same, say $m\times n$.
If $p\geq m$ and $q\geq n$, then we can take $$ H=\begin{bmatrix} I_{m}&0_{m\times(p-m)}\end{bmatrix}\ \ \ \ \Delta_i=\begin{bmatrix}A_i&0_{m\times(q-n)}\\0_{(p-m)\times n}&0_{(p-m)\times(q-n)}\end{bmatrix},\ \ \ \ E=\begin{bmatrix}I_n\\0_{(q-n)\times n}\end{bmatrix} $$
If $p<m$ say, we quickly run into trouble. For example let $$ A_1=\begin{bmatrix}1&2 \end{bmatrix},\ \ A_2=\begin{bmatrix}3&4\end{bmatrix}, $$ with $p=1$, $q=1$. The sizes of $H$ and $E$ are forced to be $1\times 1$ and $1\times 2$. Say we put $$\Delta_1=\begin{bmatrix}\alpha \end{bmatrix},\ \ \ \Delta_2=\begin{bmatrix}\beta \end{bmatrix}.$$
So $A_1=H\Delta_1 E$ and $A_2=H\Delta_2 E$ are $$ \begin{bmatrix}1&2 \end{bmatrix}=\begin{bmatrix} H_1 \end{bmatrix}\begin{bmatrix}\alpha \end{bmatrix}\begin{bmatrix}E_{1}&E_{2}\end{bmatrix}=\begin{bmatrix}\alpha H_1E_1&\alpha H_1E_2\end{bmatrix} $$ and $$ \begin{bmatrix}3&4 \end{bmatrix}=\begin{bmatrix} H_1 \end{bmatrix}\begin{bmatrix}\beta \end{bmatrix}\begin{bmatrix}E_{1}&E_{2}\end{bmatrix}=\begin{bmatrix}\beta H_1E_1&\beta H_1E_2\end{bmatrix}. $$ That is, we need $$ 1=\alpha H_1E_1,\ \ 3=\beta H_1E_1,\ \ 2=\alpha H_1E_2,\ \ 4=\beta H_1E_2. $$ None of the parameters can be zero. But then the first two equations give $$ \frac1\alpha=\frac3\beta, $$ while the second gives $$\frac2\alpha=\frac4\beta.$$
So no factorization exists.