0
votes
0answers
24 views

Smith normal forms and a math program

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 ...
0
votes
1answer
19 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 $D=\mathbb{Z}$.
0
votes
0answers
94 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 ...
1
vote
2answers
168 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 ...
3
votes
1answer
817 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 & ...
3
votes
1answer
134 views

Do a matrix and its transpose have the same invariant factors over a PID?

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? ...