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).
117 questions
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Smith normal form of a Jordan Matrix
I'm trying to find the Smith normal form of this $4\times4$ Jordan matrix:
$$
\begin{pmatrix}
\lambda-a&1&0&0\\
0&\lambda-a&1&0\\
0&0&\lambda-a&1\\
0&0&0&...
- 13
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0
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47
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Quotient of two subgroups of the free abelian group $\mathbb{Z}^r$ and Smith Normal Form.
Suppose we have free abelian groups $H \leq G \leq \mathbb{Z}^r$. I'd like to understand the quotient $G/H$. Suppose $\operatorname{rank}(H)=p$, $\operatorname{rank}(G)=s$ and that $(h_i)_{i=1,..,p}, (...
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Given a diagonal matrix, how can we find its smith normal form? [closed]
Suppose we have a diagonal matrix $$A={\rm diag} \{k_1,\dots,k_n\},$$
where $k_i\in \mathbb{N}^+$, is there any way to describe the smith normal form of $A$?
As far as I am concerned, the only thing ...
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Smith normal form of 3x3 matrices
Let $A=\begin{pmatrix}a & d & e\\ 0 & b & f\\ 0 & 0 & c \end{pmatrix}$ be an integer matrix such that $abc=n$, $0 \leq d < b$ and $0 \leq e,f < c$. I'm trying to find an ...
user1278517
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Classifying maps of finitely generated abelian groups up to automorphism
We have a nice characterization of f.g. abelian groups that factors them into cyclic groups. A homomorphism between them is just a matrix of maps $\mathbb{Z}_{p^i} \to \mathbb{Z}_{p^j}$ between the ...
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Finding Smith normal form of a $\mathbb C[\lambda]$-matrix
Let $J_n(\lambda)$ denote the Jordan block of size $n$ with eigenvalue $\lambda$, i.e.
$$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 ...
- 49
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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 ...
- 444
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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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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 ...
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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
$$H = \begin{bmatrix} x & y \\ 0 & z \\ 0 & 0 \end{...
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Smith Normal forms in a polynomial ring
Let $M$ be a $\mathbb{C[x]}$ module with generators $m_1,m_2$ and relations : $$(x^2+ix)m_1+(x+i)m_2=0 \\ (-2x+2i)m_1+ (x^2+1)m_2=0 $$
Find integers $t,n_1,...,n_s \in \mathbb{N_0}$ and $\lambda_1,\...
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Find Smith's normal form of $\begin{pmatrix} 7\theta-14 & \theta^2/2-2\theta+3\\ \theta^2 - 4\theta + 6 & 0 \end{pmatrix}$
I need to find the Smith Normal form of
$\begin{pmatrix}
7\theta-14 & \theta^2/2-2\theta+3\\
\theta^2 - 4\theta + 6 & 0
\end{pmatrix}$. This is a sub problem of a bigger problem. The problem ...
- 753
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Smith-McMillan form of matrix of transfer functions
I have the following matrix of transfer functions for the linear system $\dot{x}=Ax(t)+Bu(t)$, $y=Cx(t)+Du(t)$. $A$ is a 4x4 matrix, $x$ is a 4x1 vector, $B$ is a 4x2 matrix (not that any of this ...
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A stronger form of Bezout's lemma and Smith normal form
Context: I am interested in the Smith decomposition of the matrix
$$
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 = \...
- 3,460
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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 ...
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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 ...
- 3,000
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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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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 ...
user965463
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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 \...
- 1,604
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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*}...
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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 ...
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A bit of trouble computing the Smith normal form of a matrix?
I am trying to diagonalize the following $\lambda-$matrix:
$$\left(
\begin{array}{cccc}
-\lambda -16 & -17 & 87 & -108 \\
8 & 9-\lambda & -42 & 54 \\
-3 & -3 & 16-\...
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Smith normal form and gcd?
I am trying to understand how to reduce different matrices to Smith normal form. I have tried to read online and in our textbook, but I do not quite understand the explanations and general proofs. In ...
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Smith normal form of matrix over $\Bbb Z$?
I was wondering if someone could help me find the Smith normal form of the matrix A over $\Bbb Z$ defined as follows:
$$A =
\begin{bmatrix}
1 & 1 & 1 & 1 & 1\\
1 & 2 & 4 & ...
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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 ...
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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 $MM^T$?
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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:
$$\left(\begin{array}{rrr}
6 & 18 & 15 \\
12 & 8 & 9 \\
10 & 6 & 8 ...
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Row reducing an integer matrix
Given a $n\times n$ integer matrix, what is the best row reduction that can be found using only integer row operations of the form:
an integer multiple of row $i$ can be added to row $j$
row i can be ...
- 289
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Relationship between Hermite Normal Form and Smith decomposition
Consider a $2\times2$ matrix $P$ with entries in $\mathbb{Z}$ and $\det(P)=N$. Its (row-wise, lower) Hermite Normal Form is given by
$$
H=\begin{pmatrix} d & 0 \\ s & \bar{d}\equiv N/d\end{...
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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
...
- 271
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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\\...
- 271
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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 $(...
- 1,946
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657
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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,...,...
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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 $M_{(n+1) \times (n+1)} = \left[
\begin{array}{c |c}
A & \begin{matrix} 0 \\ 0 \\...
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How can I obtain the Smith normal form of large matrix?
Sorry that I could not include the matrix to the title, it goes over the limit of character number.
The given matrix is \begin{bmatrix}x-\frac{1}{2}&\frac{1}{2}&\frac{1}{2}&-\frac{1}{2}\\-\...
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$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 ...
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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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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 =
\...
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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 ...
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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 ...
user889696
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1
answer
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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 ...
user889696