Questions tagged [smith-normal-form]
For questions related to Smith normal form. It is a normal form that can be defined for any matrix (not necessarily square) with entries in a principal ideal domain (PID).
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Solve system of linear equations modulo n
Suppose I have a positive integer $n$ and a system of linear equations over $n$, i.e., $Mx \equiv c \pmod n$ where $M$ is a matrix and $c$ is a vector. Given $M,c,n$, is it possible to solve this ...
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Classifying maps of finitely generated abelian groups up to automorphism
We have a nice characterization of f.g. abelian groups that factors them into cyclic groups. A homomorphism between them is just a matrix of maps $\mathbb{Z}_{p^i} \to \mathbb{Z}_{p^j}$ between the ...
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Finding Smith normal form of a $\mathbb C[\lambda]$-matrix
Let $J_n(\lambda)$ denote the Jordan block of size $n$ with eigenvalue $\lambda$, i.e.
$$J_n(\lambda)=\begin{pmatrix}
\lambda & 1 & & \\
& \lambda & \ddots & \\
& & \...
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Should I add an interesting conculsion from my research to wikipedia? [closed]
The stated phrase in the title is a bit negatively misguided. For context: in my amateur research, I have come upon the conclusion that
$$SNF(A \times B) = SNF(A)\times SNF(B)$$
Where SNF(X) denotes ...
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Why Smith normal form gives isomorphic modules?
I have an answer to the problem but I use some (trivial) diagram chasing by $5$-Lemma. Consider a principle ideal domain $A$ and a finitely generated module $M$ over $A$. Since $A$ is Noetherian, we ...
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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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Help determining the rank of a module
I have the following question on my homework:
Find the rank of the subgroup of $\mathbb{Z}^3$ generated by (2,-2,0), (0,4,-4), and (5,0,-5)
I've seen the comment on this post which inspired me to ...
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Image of matrix modulo prime power
Given an integer matrix $A \in \mathbb{Z}^{m\times m}$. I know one van find the number of elements in the image of $A$ modulo $p^k$ by looking at the Smith Normal Form, i.e. $S = PAQ$ with $P$ en $Q$ ...
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invariant factors of integer matrix with parameters
Given two integer matrices $A_1$ and $A_2$. Consider the matrix $M(x_1,x_2) = x_1A_1 + x_2A_2$, where $x_1,x_2 \in \mathbb{C}$. The matrices are chosen such that $M(x_1,x_2)$ has rank at least $d$ (...
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What is meant by "invertible" matrices in the creation of a SNM
I just read up on wikipedia on the Smith Normal Matrix. But what is meant by an invertable matrix. For example if you have a start matrix with only PID values does that mean the other matrices don't ...
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Are the invertible matrices which are used to find the SNF always part of the ideal principle domain?
Let's say SNF = TAT^-1.
Do T and T inverse always only have elements part of the domain?
For example, if we have an integer matrix, will T and T inverse only have integer values (not rational numbers)....
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Solving linear equations over $\mathbb{Z}$ using Smith normal form
To solve linear equations over $\mathbb{Z}$ we have a system of linear equations represented by some integer matrix $A$ of $n \times m$ dimension and $b \in \mathbb{Z}^n$. Such that solving $Ax = b$ ...
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Smith normal form of morphisms between non-free $R$-modules
If $R$ is a ring and, further, a PID, a morphism of $f : M \to N$ of finitely generated, free $R$-modules has a Smith normal form. Does this also hold when $M$ and $N$ are finitely generated but not ...
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Help to find two sets of two linear independent vectors satisfies certain properties
I am trying to find two sets of two linear independent row vectors in $\mathbb Z^2$ satisfies certain properties, I made a program in Matlab to generate such vectors, however, it still hasn't found ...
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What does it mean for a submodule of a module over a PID to have invariant factors $1$ or $0$?
I will take a particular example for simplicity: suppose $D$ is a PID and $M$ is finitely generated submodule of $D^5$, say, with set of generators $x_i, i=1,2,3,4,5$. Suppose also that the smith ...
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