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Tagged with smith-normal-form commutative-algebra
4 questions
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Clarifications on Smith Normal Form
I'm solving an exercise where I need to find the Smith normal form of a matrix. As I understood, what I need to do for a $2\times3$ matrix is to find the determinant of each of its $1\times1$ and $2\...
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Terminology for invariant factors of quotient module over PID
Let $A$ be a PID, $M$ a finitely generated $A$-module and $N$ a submodule. By the structure theorem of finitely generated modules or by Smith normal form, $M/N \cong \prod A/(a_i)$ for certain $a_i \...
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Matrix is similar to its transpose over every field
I want to prove that every matrix is similar to its transpose. My lecturer gave us this exercise:
Let $\Bbb{F}$ be a field, $A\in M_{n\times n}$ and $A^t$ its transpose. We define $M,L=F^n$ to be $\...
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Smith normal form Zariski-locally
Let $A\in GL_n(\mathbb{C}((t)))$, i.e. some invertible matrix over the ring of Laurent series. It is known that there are $P,Q\in GL_n(\mathbb{C}[[t]])$, such that $PAQ$ is diagonal. This is just ...
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