Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [smith-normal-form]

The tag has no usage guidance.

0
votes
1answer
29 views

Smith normal form of the following matrix

Let $$ A = \begin{bmatrix} 66 & 30\\ 12 & 4 \end{bmatrix}$$ I've been trying to find the smith normal form of this matrix, and I keep getting the wrong answer. Here are my workings; gcd of ...
1
vote
3answers
38 views

Find the structure of $ \mathbb Z ^{3} / K $ with $K$ the image of a matrix

I have this matrix: $$ A= \begin{pmatrix} 2 & 5 & -1 & 2\\ -2 & -16 & -4 & 4 \\ -2 &-2 &0 &6 \end{pmatrix} $$ If we set K as the Image of this matrix, how do you ...
1
vote
0answers
24 views

Smith Normal Form of a companion matrix of monic polynomial

Let $C(f)$ be the companion matrix of a monic polynomial $f(t)\in \mathbb{F}[t]$. I need to show that the Smith Normal Form of $tI - C(f)$ is equal to the diagonal matrix $\,diag(1,1,1,...,f(t))$. A ...
0
votes
0answers
14 views

Is the product of two Smith Normal Forms a the Smith normal form of the product?

Suppose A and B are square matrices of the same size over a PID R. Does the Smith Normal form of AB equal the product of the Smith normal form of A and B? I think this should be false. However, I can'...
1
vote
1answer
64 views

Let $G=\mathbb{Z}/24\mathbb{Z}\times\mathbb{Z}/6\mathbb{Z}\times\mathbb{Z}/3\mathbb{Z}$. Consider the quotient group $H=G/\langle (10,3,2)\rangle$.

Determine a direct product of cyclic groups that is isomorphic to $H$. The Smith Normal form can be used to find the invariant factors in the structure theorem for finitely generated abelian groups. ...
1
vote
1answer
42 views

Structure Theorem for finitely generated Modules over a PID, Decomposing an Example Problem and finding Bases

I came across this Problem in Terms of my exam preparation: a.) Let N $\subset \mathbb{Z}^3$ be the submodule generated by the set {(2,4,1),(2,-1,1)}. Find a Basis {$f_1,f_2,f_3$} for $\mathbb{Z}^3$, ...
5
votes
1answer
151 views

Smiths normal form is similar to $xI-A.$

I am reading Rational Canonical form from The Abstract Algebra book by Dummit and Foote. I have some doubt in Smith normal form. Smiths normal for says for any $n\times n$ square matrix $A$ over an ...
-2
votes
1answer
47 views

Smith transformation [duplicate]

How to transfer the following matrix into Smith normal form? $$\left[\begin{matrix} 2 & -2b & 0 \\ 0 & 2 & -2c \\ -2a & 0 & 2 \end{matrix}\right]$$ The final answer ...
0
votes
0answers
22 views

Determine the quotient $\mathbb{Z}^3\big/ A\mathbb{Z}^3$ with Smith normal form

I found the Smith normal form of a matrix $A$ to be $$B=\begin{pmatrix} 1 & 0 &0\\ 0 & 4 & 0\\ 0&0& 8 \end{pmatrix}$$ Can I conclude now that $\mathbb{Z}^3\big/ A\mathbb{Z}^3=...
0
votes
0answers
20 views

Smith Normal Form when characteristic poly=minimal poly.

I have the foreknowledge that the following matrix has the same minimal and characteristic polynomial: $$A=\begin{pmatrix}1 & 1 & 0 & 0\\ -1 & -1 & 0 & 0\\ -2 & -2 & 2 ...
2
votes
1answer
75 views

Calculate Smith normal form, cyclic group decomposition

Can someone please check my working for the following problem? Let $A$ be the abelian group generated by elements $x,y,z$ with relations $7x+5y+2z=0, 3x+3y=0, 13x+11y+2z=0$. Decompose $A$ as a ...
3
votes
1answer
137 views

Smith Normal Form

I would like to put this matrix below into Smith Normal Form over $\mathbb{Q}[x]: $ $$\left( \begin{array}{ccc} 7 & x & 0 & -x \\ 0 & x-3 & 0 & 3\\ 0 & 0 & x-4 & 0 \...
2
votes
2answers
596 views

Z-Smith Normal Form

I am a bit confused on how to put the matrix \begin{bmatrix}4&2&4\\3&3&4\\2&2&2\end{bmatrix} in $\mathbb{Z}$-Smith Normal Form. I know this can be done using unimodular ...
2
votes
2answers
303 views

Reducing a matrix to Smith normal form

I am trying to reduce the following matrix to Smith normal form $$A= \begin{pmatrix} 1&0&0\\ 1&2&0\\ 1&0&3 \end{pmatrix}$$ Whatever row and column operations I try, I end up ...
3
votes
1answer
1k views

Reduce matrix to Smith Normal form.

I've been given the finitely generated abelian group: $$\langle x_1, x_2 \mid 6x_1-6x_2, -6x_1-12x_2, 4x_1-8x_2\rangle$$ and written the corresponding matrix: $$A=\begin{pmatrix} 6 & -6 \\ -6 &...
6
votes
1answer
351 views

Problem with Smith normal form over a PID that is not an Euclidean domain

This is an homework exercise of the Algebra lecture. I need to evaluate the Smith normal form of the following matrix $$A:=\begin{pmatrix}1 & -\xi & \xi-1\\2 \xi&8&8\xi+7\\\xi&...
1
vote
0answers
64 views

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 ...
4
votes
1answer
1k views

Determining the Smith Normal Form

Consider the integral matrix $$R = \left(\begin{matrix} 2 & 4 & 6 & -8 \\ 1 & 3 & 2 & -1 \\ 1 & 1 & 4 & -1 \\ 1 & 1 & 2 & 5 \end{matrix}\right).$$ ...
21
votes
1answer
12k views

Computing the Smith Normal Form

Let $A_R$ be the finitely generated abelian group, determined by the relation-matrix $$R := \begin{bmatrix} -6 & 111 & -36 & 6\\ 5 & -672 & 210 & 74\\ 0 & -255 &...