# Tagged Questions

I am interested to know the Smith normal form of $4 \times 2$ matrices $M$: The two cases of my interests are: (1). $$M_1= \begin{pmatrix} 3 & 0\\ -5 & 4\\ 4 & -5\\ 0 & 3 ... 1answer 21 views ### Are rank and determinantal rank the same over a PID? Are the notions of rank and determinantal rank equivalent for an m\times n matrix A with entries in a principal ideal domain D? I'm specifically interested in the case D=\mathbb{Z}. 0answers 95 views ### Homology out of Smith normal form Let R be a PID and A: R^m\rightarrow R^n and B:R^n\rightarrow R^o with BA=0 and Smith normal forms A=P\mathrm{diag}(a_1,\ldots,a_r,0,\ldots,0)Q^{-1} and ... 2answers 170 views ### surjective homomorphism between principal ideal rings I asked this question before but did not get an answer, I tried solving it myself and I think I'm heading somewhere, I just need a push in the right direction. Let \phi: R \to S be a homomorphism ... 1answer 830 views ### How do I get this matrix in Smith Normal Form? And, is Smith Normal Form unique? As part of a larger problem, I want to compute the Smith Normal Form of xI-B over \mathbb{Q}[x] where$$ B=\begin{pmatrix} 5 & 2 & -8 & -8 \\ -6 & -3 & 8 & 8 \\ -3 & ...
I suspect this is true since it holds in the case over a field. But suppose $A\in M_{m\times n}(R)$ where $R$ is a PID. Does it still hold that $A$ and $A^{T}$ have the same invariant factors? ...