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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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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 & \\ & & \...
Cyankite's user avatar
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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 ...
IV-301's user avatar
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1 answer
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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 ...
user108580's user avatar
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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\...
WittyCatchphrase's user avatar
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1 answer
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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 ...
modz's user avatar
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0 answers
35 views

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 $Q$ ...
MatthysJ's user avatar
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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 $d$ (...
MatthysJ's user avatar
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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 ...
IV-301's user avatar
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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)....
IV-301's user avatar
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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 $Ax = b$ ...
Txim's user avatar
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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 ...
richokicked800goals's user avatar
2 votes
1 answer
164 views

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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2 votes
1 answer
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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 ...
Victor's user avatar
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1 answer
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Find the Smith normal form of certain matrix

I need to find all invariant factor of matrix $\begin{pmatrix} \lambda +1 & 2 & -1 \\ 1 & \lambda & -3 \\ 1 & 1& \lambda-4 \end{pmatrix} = A(\lambda)$ that is, by existence of ...
Victor's user avatar
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Do equivalent matrices have the same image

Let say we have R-module homomorphisms $T_1 ,T_2 : R^m \rightarrow R^n $ with matrices $A$ and $B$ respectively. If $B$ can be obtained from $A$ by just performing elementary column and row operations,...
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HNF of $\begin{bmatrix}c & 0 \\ 0 & c \\ a & b \end{bmatrix}$

Is there any way to find $x, y, z \in \mathbb{Z}$ such that, given $a, b, c \in \mathbb{Z},\ a, b, c > 0,\ (a, b, c) = 1$ matrix $$H = \begin{bmatrix} x & y \\ 0 & z \\ 0 & 0 \end{...
Andrey's user avatar
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1 answer
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Smith Normal forms in a polynomial ring

Let $M$ be a $\mathbb{C[x]}$ module with generators $m_1,m_2$ and relations : $$(x^2+ix)m_1+(x+i)m_2=0 \\ (-2x+2i)m_1+ (x^2+1)m_2=0 $$ Find integers $t,n_1,...,n_s \in \mathbb{N_0}$ and $\lambda_1,\...
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Find Smith's normal form of $\begin{pmatrix} 7\theta-14 & \theta^2/2-2\theta+3\\ \theta^2 - 4\theta + 6 & 0 \end{pmatrix}$

I need to find the Smith Normal form of $\begin{pmatrix} 7\theta-14 & \theta^2/2-2\theta+3\\ \theta^2 - 4\theta + 6 & 0 \end{pmatrix}$. This is a sub problem of a bigger problem. The problem ...
H-a-y-K's user avatar
  • 701
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0 answers
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Smith-McMillan form of matrix of transfer functions

I have the following matrix of transfer functions for the linear system $\dot{x}=Ax(t)+Bu(t)$, $y=Cx(t)+Du(t)$. $A$ is a 4x4 matrix, $x$ is a 4x1 vector, $B$ is a 4x2 matrix (not that any of this ...
Programmer's user avatar
1 vote
0 answers
670 views

A stronger form of Bezout's lemma and Smith normal form

Context: I am interested in the Smith decomposition of the matrix $$ A = \begin{pmatrix} a & b\\ 0 & c \end{pmatrix}~, $$where $a,b,c\in\mathbb Z$. I know that the Smith normal form is $$ S = \...
Pranay's user avatar
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Intersection of the null space of a matrix with integer vectors

I have an $m\times n$ rational matrix $A$ which is full rank and has linearly independent columns. With $m>n$ I would like to find all integer values of $x$ for which $$A\vec x=0$$ If $A$ was an ...
Cameron's user avatar
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1 answer
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Question regarding divisibility of invariant factors in Smith Normal Form.

I am trying to understand the algorithm presented in Wikipedia on how to calculate the Smith Normal Form of a matrix. I understand how to transform the matrix into a diagonal matrix and that the ...
Jhon Doe's user avatar
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1 answer
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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 \...
Bart Michels's user avatar
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1 vote
1 answer
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How is the matrix P is found and why using Smith Normal Form?

Here is the question I am trying to solve from Allen Hatcher's book: Compute the simplicial homology groups of the Klein bottle using the $\Delta$-complex structure described at the beginning of this ...
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$A^T$ has the same Smith normal form as $A$

Let $A\in R^{n\times m}$ be a matrix over a PID $R$. Show that $A$ and its transpose $A^T$ have the same Smith normal form, meaning that the ideals $\{0_R\}\subsetneq \langle d_r \rangle \subseteq \...
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1 answer
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Calculate Smith Normal Form

Let \begin{align*} A=\begin{pmatrix} 1&0&-1&2\\ 1&2&1&0\\ 1&0&2&2\\ 1&2&2&0 \end{pmatrix} \in \mathbb{Z}^{4\times 4} \end{align*}...
Quotenbanane's user avatar
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1 vote
0 answers
230 views

Computing Smith normal form of matrix of integers

I known that the Smith normal form of $A$ provides two unimodular matrices $U$ and $V$ of respective dimensions $m \times m$ and $n \times n$ such that the matrix $$B=[b_{i,j}]=UAV$$ and B has the ...
lovemath's user avatar
0 votes
2 answers
156 views

A bit of trouble computing the Smith normal form of a matrix?

I am trying to diagonalize the following $\lambda-$matrix: $$\left( \begin{array}{cccc} -\lambda -16 & -17 & 87 & -108 \\ 8 & 9-\lambda & -42 & 54 \\ -3 & -3 & 16-\...
Red Banana's user avatar
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Smith normal form and gcd?

I am trying to understand how to reduce different matrices to Smith normal form. I have tried to read online and in our textbook, but I do not quite understand the explanations and general proofs. In ...
Rory's user avatar
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1 vote
2 answers
190 views

Smith normal form of matrix over $\Bbb Z$?

I was wondering if someone could help me find the Smith normal form of the matrix A over $\Bbb Z$ defined as follows: $$A = \begin{bmatrix} 1 & 1 & 1 & 1 & 1\\ 1 & 2 & 4 & ...
Rory's user avatar
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0 answers
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Is there an algorithm to get a set of $n$ distinct vectors which generates the given $n$ vectors, whose degree follows the irreducible factorization?

For example, for $A=\begin{pmatrix}2&1\\0&2\end{pmatrix}$, its smith normal form is $\begin{pmatrix}1&0\\0&4\end{pmatrix}$. So I can calculate the smith normal form, or the irreducible ...
user5876164's user avatar
0 votes
1 answer
82 views

Smith Normal Form of the product of a matrix with its transpose after swapping columns

Suppose I have an integer matrix $M$ and I consider the Smith Normal Form of the matrix $MM^T$. If I then swap two columns of $M$, does that affect the Smith Normal Form of $MM^T$?
astro117's user avatar
1 vote
1 answer
82 views

What does this step in the following matrix algorithm mean?

I am trying for hours to calculate in every step the Smith Normal form of this matrix but without success: $$\left(\begin{array}{rrr} 6 & 18 & 15 \\ 12 & 8 & 9 \\ 10 & 6 & 8 ...
Annalisa's user avatar
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0 answers
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Row reducing an integer matrix

Given a $n\times n$ integer matrix, what is the best row reduction that can be found using only integer row operations of the form: an integer multiple of row $i$ can be added to row $j$ row i can be ...
Cameron's user avatar
  • 429
1 vote
1 answer
308 views

Relationship between Hermite Normal Form and Smith decomposition

Consider a $2\times2$ matrix $P$ with entries in $\mathbb{Z}$ and $\det(P)=N$. Its (row-wise, lower) Hermite Normal Form is given by $$ H=\begin{pmatrix} d & 0 \\ s & \bar{d}\equiv N/d\end{...
BeMuSeD's user avatar
  • 105
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0 answers
244 views

Smith Normal Form of polynomial matrix

Here is a question on finding the Smith Normal Form of a polynomial matrix Smith normal form of a polynomial matrix I am wondering why my method is wrong here: $\begin{bmatrix} x^2&x-1\\ x&x^2 ...
Nicky's user avatar
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1 vote
0 answers
43 views

Calculate Smith Normal Form of matrix

I am struggling to calculate the Smith Normal Form of this matrix. I know it is wrong because I checked on a computer. Can someone help me where I am going wrong? $\begin{bmatrix} 6&2&3&0\\...
Nicky's user avatar
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3 votes
1 answer
58 views

Non-cyclic Finite abelian group generated by 2 elements

I've thinking for a couple hours but I were not able to figure the following question: Suppose I have a non-cyclic finite abelian group generated by two elements $a$ and $b$. Is it possible to have $(...
Alejandro Tolcachier's user avatar
1 vote
2 answers
575 views

Generalized Jordan canonical form

Suppose $\mathbb F_q$ is the finite field of order $q$. Let $f(x)=x^d-a_{d-1}x^{d-1}-\cdots-a_{1}x-a_0\in\mathbb F_q[x]$ be irreducible with $\deg (f(x))=d$. Prove that we can find a basis $\{e_1,...,...
Bach's user avatar
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Connections between a submatrix and a partitioned matrix.

I am thinkng about some relations between a symmetric matrix $A_{n \times n}$ and another symmetric matrix $M_{(n+1) \times (n+1)} = \left[ \begin{array}{c |c} A & \begin{matrix} 0 \\ 0 \\...
integer_wise's user avatar
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1 answer
68 views

How can I obtain the Smith normal form of large matrix?

Sorry that I could not include the matrix to the title, it goes over the limit of character number. The given matrix is \begin{bmatrix}x-\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&-\frac{1}{2}\\-\...
okw1124's user avatar
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2 votes
0 answers
95 views

$F[x]$-modules isomorphism

Let $\mathbb{F}$ be a field, and $A\in M_{n\times n}(\mathbb{F})$. Define $M,L=\mathbb F^n$ to be $\mathbb {F}[x]$-modules, s.t. for every $m\in M,l\in L$ : $f(x)m=f(A)m$ $f(x)l=f(A^t)l$ Prove that ...
DBXz's user avatar
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1 vote
1 answer
387 views

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 $\...
Math101's user avatar
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1 vote
1 answer
396 views

Using Smith Normal Form to understand cokernel of a map between $\mathbb{Z}$-modules

I want to explicitly understand the $\mathbb{Z}$-module I constructed as $M = \mathbb{Z}^4/\mathrm{im}(A)$, where $A\colon \mathbb{Z}^6 \to \mathbb{Z}^4$ is represented by the matrix $$ A = \...
Sascha Baer's user avatar
1 vote
1 answer
99 views

Why the diagonal elements of the Smith normal form of a boundary matrix are the torsion coefficients of a homology module?

Can you help in proving the isomorphism going between the torsion of $H_p$ and $\left(\bigoplus_{i} R / d_{p i} R\right)$? Where $H_p$ is the p-th homolgy module, R is a commutative PID, and $d_{i}$'s ...
HNB's user avatar
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1 vote
0 answers
182 views

Module characteristic polynomial.

I know that if I have a matrix $A,$ the characteristic polynomial is determinant of the matrix $(A-\lambda I)$ where, $\lambda$ is an eigenvalue and $I$ is an identity matrix, and the characteristic ...
user avatar
1 vote
1 answer
364 views

Module elementary divisors.

I am practicing for myself how to find all possible elementary divisors and the corresponding invariant factors for an $R$-module of order $(x-1)^3(x +1)^2$ where $R = k[x]$ and $k$ is a field. But ...
user avatar
1 vote
0 answers
22 views

Taking quotients over isometric groups and determining the cokernel of transformations

We know that given group $A$ and subgroups $B_1,B_2$ such that $B_1\cong B_2$, it is not necessary that $A/B_1\cong A/B_2$. Now take for instance the following transformation $\varphi:\mathbb{Z}^3\...
Emma Amabilis's user avatar
1 vote
1 answer
112 views

If $f:\mathbb{Z}^m\to\mathbb{Z}^m$ is a module homomorphism, then $\mathbb{Z}^m/\operatorname{im}(f)$ is a finite abelian group

If $f:\mathbb{Z}^m\to\mathbb{Z}^m$ is an injective group homomorphism, then is $\mathbb{Z}^m/\operatorname{im}(f)$ a finite abelian group? I think yes. Let the homomorphism be given by a $m\times m$ ...
vidyarthi's user avatar
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1 answer
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Smith normal form and elementary divisors

Say I have the matrix $\begin{pmatrix}16&16&8\\8&6&2\\3&4&2\end{pmatrix}$. To give a structure for a module homomorphism whose representation is given by this matrix, I must ...
vidyarthi's user avatar
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