Given an matrix $ K \in GF(2)$, of dimension $m\times n$ ($m>n$) we can write its Smith normal form as $$ A = P K Q $$ where $A = \mathrm{diag}(a_1,a_2,\dots,0,0)$ (here $a_1=a_2=\dots=1$, since we are in $GF(2)$). Playing around with some examples it appears that $(Q A^T P) K = I_n$.
I followed the idea behind SVD and the pseudoinverse, and it seems to work, but I cannot formally convince myself that's the case. Am I fooling myself with some (too) simple examples?
Is this Correct?
We are looking for a matrix $X$ such that $XK=I_n$, so $X$ has dimensions $n\times m$.
If we choose $X = QA'P$ then we have $K = P^{-1} A Q^{-1}$ from the previous equation, and thus we have $$ XL = QA'PP^{-1}AQ^{-1} = QA'AQ^{-1} = Q Q^{-1} = I_n $$ Since $A'A = I_n$. Do we know that $P$ and $Q$ are in $GF(2)$?