Suppose I have some matrix $M$ with entries from a PID. Define the column rank and row rank as you would expect: the rank of the module generated by the columns (resp. rows).
Is it always true that the column rank is equal to the row rank?
linear algebramatrix-rankmodules
Suppose I have some matrix $M$ with entries from a PID. Define the column rank and row rank as you would expect: the rank of the module generated by the columns (resp. rows).
Is it always true that the column rank is equal to the row rank?
Best Answer
Yes, due to the existence/uniqueness of Smith normal form:
Equivalence of $A$ and $B$ here means there are invertible square matrices $P\in M_n(R)$ and $Q\in M_m(R)$ with $B=PAQ$. In particular the rank of the images of $A$ and $B$ agree, and similarly for the null spaces.