My 2 Questions:
Any matrix can be brought into block form however, why does $A$ and $B$ need to have the same rank?
It follows further that Any $m \times n$ matrix is equivalent to the block matrix where the Identity submatrix in the top left is $s \times s$ where $s = \mbox{rank}(A)$. Why is this the case??
My thoughts:
Is it due to the fact that $\mbox{rank} (A)$ the maximum number of linearly independent column/row vectors in the matrix $A$?
What I believe to be the definition- Let $A$ and $B$ be matrices over the field $\Bbb K$. If both $A$ and $B$ have rank $s$, then they can both be brought into the same block (column and row reduced) form.
Say this number were $n$. Then this implies that there are n linearly independent column vectors in $A$. If the rank of $B$ was not $n$, but m then would the size of the identity matrix in the top left of its block form be different?
The block form I am referring to is the following: $$ \left( \begin{array}{c|c} I_s & 0_{s,n-s} \\ \hline 0_{m-s,s} & 0_{m-s,n-s} \end{array} \right) $$