# Finding good bases to represent any rectangular matrix as a block matrix with identity submatrix

This question is a generalization of Finding bases such that the matrix representation is a block matrix where one submatrix is the identity matrix .

### Question

For any linear map $$L:\mathbb{R}^n \to \mathbb{R}^m$$ where $$n\neq m$$,
given its matrix representation $$[L]^{\mathcal{E}_n}_{\mathcal{E}_m}$$, say $$\begin{pmatrix}a_{1,1} & \dots & a_{1,n} \\ \vdots & \ddots & \vdots \\ a_{m,1} & \dots & a_{m,n}\end{pmatrix}$$, with respect to the standard basis $$\mathcal{E}_n$$ of $$\mathbb{R}^n$$ and $$\mathcal{E}_m$$ of $$\mathbb{R}^m$$,
must we be able to find basis $$\alpha$$ for $$\mathbb{R}^n$$ and $$\beta$$ for $$\mathbb{R}^m$$ such that $$[L]^{\alpha}_{\beta} = \begin{pmatrix}\mathbf{I}_{r} & \mathbf{O} \\ \mathbf{O}& \mathbf{O} \end{pmatrix}$$,
where $$\mathbf{I}_{r}$$ is an $$r\times r$$ identity matrix with $$r=\text{Rank}(L)$$, and $$\mathbf{O}$$'s are some zero matrices?
If yes, what are the systematic ways (if any) to find it?

### Thoughts

My professor casually said that it is true and left it as an exercise, giving hints along the lines of "do row / column operations to get the change of basis matrices".
It was used in subsequent proofs in the class so probably it is really true.

The closest I know / can find (which are more sophisticated than "just" row / column operations) are

1. Diagonalization, which is for $$n=m$$ and the diagonal entries are eigenvalues, and
2. Singular Value Decomposition, which is for $$n\neq m$$ but still gives $$\begin{pmatrix}\mathbf{D} & \mathbf{O} \\ \mathbf{O}& \mathbf{O} \end{pmatrix}$$ only where $$\mathbf{D}$$ is a diagonal matrix.

$$\begin{pmatrix}\mathbf{I}_{r} & \mathbf{O} \\ \mathbf{O}& \mathbf{O} \end{pmatrix}$$ sounds too good to be true... (but I am still a beginner in Linear Algebra)
I wonder if some more conditions are needed?

I also tried a bunch of keywords in Google but could not find anything.
(are there names for "a block matrix with identity submatrix"?)
I apologize if my question is not phrased in the standard way.
I would appreciate if there are some pointers.

Each Gaussian 'row move' can be represented by an elementary row matrix; similarly for column moves. Hence applying the Gaussian row/column operations is effectively the same as $$E_rAE_c = \begin{pmatrix}I&O\\O&O\end{pmatrix}=:I'$$ where $$E_r=E_1\cdots E_k$$ is the product of the row operations applied to $$A$$. So taking their inverses gives $$A=E_r^{-1}I'E_c^{-1}$$ as required.
If SVD is known for $$A$$, that is, $$A=UDV^\top$$ (with $$U,V$$ square), then write $$D=\begin{pmatrix}P&O\\O&O\end{pmatrix}=\begin{pmatrix}R&O\\O&I\end{pmatrix}\begin{pmatrix}I&O\\O&O\end{pmatrix}\begin{pmatrix}R&O\\O&I\end{pmatrix}=R'I'R'$$ where $$P$$ is a diagonal matrix of strictly positive numbers $$\sigma>0$$ and $$R$$ is also diagonal consisting of their square roots $$\sqrt{\sigma}$$. Then $$A=(UR')I'(R'V)^\top.$$