If I have a linearly independent set of vectors say
\begin{bmatrix}0&0&9&0&0&0&0&0\\1&2&0&0&0&2&0&0\\0&0&0&0&20&0&13&0\\0&20&0&20&20&18&4&0\\0&4&0&8&5&0&0&19\\0&20&0&12&9&7&0&5\\0&4&0&0&0&0&0&0\\0&0&0&0&0&11&0&6\end{bmatrix}
is it possible to use row reduction to convert the matrix into two non zero sub blocks of size 4x4 say to something like this.
\begin{bmatrix}a&b&c&d&0&0&0&0\\e&f&g&h&0&0&0&0\\i&j&k&l&0&0&0&0\\m&n&o&p&0&0&0&0\\0&0&0&0&q&r&s&t\\0&0&0&0&u&v&w&x\\0&0&0&0&y&z&a_2&b_2\\0&0&0&0&c_2&d_2&e_2&f_2\end{bmatrix}
I have tried to use row reduction by I keep ended up with 1 or 2 non zero entries where a zero should be.
The matrix is not necessarily sparse and could be dense.