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Questions tagged [smith-normal-form]

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Smith normal form of rectangular matrix in MATLAB

Suppose I've got a nonsquare integer matrix, say $\begin{pmatrix}3 & 1 & 1 & 1\\1 & 1 & 1 &1\end{pmatrix}$ and want to compute its Smith normal form--in this case, $\begin{...
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1answer
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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 ...
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3answers
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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 ...
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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 ...
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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'...
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1answer
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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. ...
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1answer
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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 $\mathbb{Z}^3$, ...
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1answer
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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 ...
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1answer
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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 ...
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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=...
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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 ...
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1answer
80 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 ...
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2answers
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How to create the Smith Mcmillan form of a polynomial matrix?

Here is the matrix: $$ G(s) = \begin{bmatrix} \frac{4}{(s+1)(s+2)} & \frac{-0.5}{s+1} \\ \frac{1}{s+2} & \frac{2}{(s+1)(s+2)}\\ \end{bmatrix} $$ And this should be the final ...
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1answer
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Finding Jordan normal form using Smith normal form

An endomorphism $T:V\to V$ of a finite dimensional $\mathbb{C}$-vector space endows $V$ with a $\mathbb{C}[X]$-module structure defined by $X\cdot v = T(v)$. From the Structure theorem for finitely ...
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2answers
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Smith Normal Form of this polynomial non square matrix in $\mathbb Q[t]$

Given $$A=\begin{bmatrix}t^3-t^2&t^2-t\\2t^3&t^4-t\\t&2t^3-2t^2\end{bmatrix} \in M_{3\times 2}(\mathbb Q[t])$$ I'm looking for the matrices $S,P,Q$ such that $S=QAP$, where $Q$ are the ...
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1answer
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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 \...
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2answers
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Finding Smith normal form of matrix over $ \mathbb{R} [ X ]$

I am trying to find the Smith normal form of matrix over $ \mathbb{R} [ X ]$ of the 4x4 matrix $$M =\begin{pmatrix} 2X-1 & X & X-1 & 1\\ X & 0 & 1 & 0 \\ 0 & 1 & X &...
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1answer
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Question on Smith normal form and isomorphism

Put $A=\begin{pmatrix} 1 & -5 & 4\\ 1 & -2 & 13\\ -2 & 13 & 7 \end{pmatrix}.$ The smith normal form of this matrix is \begin{pmatrix} 1 & 0 & 0\\ 0 & 3 & ...
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1answer
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Confusion with Smith normal form and rational canonical form.

I am taking an abstract algebra course and am getting quite confused with the terminology of invariant factors, elementary divisors, and the normal forms. I am asked to compute the rational canonical ...
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1answer
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Finding the Smith Normal Form of an integer matrix

I want to put the following integer matrix into Smith Normal Form: $$\begin{pmatrix} -9 & 6 \\ 5 & -2 \\ 6 & 3 \end{pmatrix}$$ I have done this and found the answer to be $$\begin{...
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2answers
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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 ...
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2answers
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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 ...
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1answer
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Uniqueness of Smith normal form in Z (ring of integers)

It is a very well known fact that Smith Normal Form has proven useful when dealing with the development of the structure theorem of finitely generated abelian groups. In this context, there is an ...
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Application to Smith And Hermite Normal Forms

Given $ A $ a $ n\times n $ integer matrix and $ (d_{1},...,d_{n}) $ the diagonal of its Smith Normal Form, I would like to prove that $ \mathbb{Z}^n/Im_{\mathbb{Z}}A\simeq \bigoplus_{i=1}^{n}(\...
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1answer
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Structure theorem (PIDs) from Smith Normal Form

How exactly does the structure theorem follow from Smith Normal Form? (Wikipedia statement) It is said that a presentation (map from relations to generators) is put into Smith Normal form. Now, I see ...
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Computing simplicial homology via Smith Normal Form over Rings

I am not sure whether this is the right forum to ask such a question, if not please let me know. In the context of my masters thesis, I am working on writing a program to compute simplicial homology ...
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1answer
220 views

Smith Normal Forms over Fields and PIDs

I need to reduce the following matrices into the Smith Normal form over the field $(\mathbb{Z}/2\mathbb{Z})[x]$: $$M_{1} = \left ( \begin{array}{ccc} x & 1 & 0 \\ 0 & x & 1 \\ 0 &...
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1answer
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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 &...
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1answer
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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&...
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2answers
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Smith Normal Form and classification of factor groups according to the theorem of finitely generated abelian groups

With reference to this question, it was mentioned in the comments that these problems could be solved using the Smith Normal Form. However, I am unable to extract an exact method from the Wikipedia ...
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1answer
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Find the Smith Normal Form of xI - A

So for a homework in my Abstract Algebra class we are to find the invariant factors of a matrix using the Smith Normal Form. The matrix is rather large, so I was trying a 3x3 sub matrix first to just ...
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Proof of Smith Normal Form theorem

It is a homework asking me to prove the Smith Normal Form theorem, stating that If $R$ is a PID, and $A$ is a $n\times n$ matrix over $R$, then there exists invertible matrices $P, Q$ such that $...
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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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2answers
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Computing invariant factors from Smith normal form

The goal is to find the Jordan Canonical Form of the matrix $$A=\begin{bmatrix}2&1&1&2\\0&2&0&1\\0&0&2&-1\\0&0&0&1\end{bmatrix}$$ Since the matrix is ...
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2answers
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Smith Normal Form for PIDs

Could someone provide a good reference to look up the existence and uniqueness of Smith Normal Form (SNF) for a PID? I have seen it done for Euclidean domains but not for a PIDs. I know the ...
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1answer
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Smith normal form of a polynomial matrix

I have the following matrix $$P(s) := \begin{bmatrix} s^2 & s-1 \\ s & s^2 \end{bmatrix}$$ How does one compute the Smith normal form of this matrix? I can't quite grasp the algorithm.
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1answer
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Determining the Smith Normal Form

Consider the integral matrix $$R = \left(\begin{matrix} 2 & 4 & 6 & -8 \\ 1 & 3 & 2 & -1 \\ 1 & 1 & 4 & -1 \\ 1 & 1 & 2 & 5 \end{matrix}\right).$$ ...
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1answer
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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 &...
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1answer
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Smith Normal Form

Would the Smith Normal Form of the following matrix over $\mathbb Q[x]$ $$\begin{pmatrix}   (x+a)(x+b) & 0 & 0 &0 \\  0 & (x+c)(x+d) & 0 & 0 \\   0 ...
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1answer
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How do I get this matrix in Smith Normal Form? And, is Smith Normal Form unique?

As part of a larger problem, I want to compute the Smith Normal Form of $xI-B$ over $\mathbb{Q}[x]$ where $$ B=\begin{pmatrix} 5 & 2 & -8 & -8 \\ -6 & -3 & 8 & 8 \\ -3 & -...
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Finding the correct pivot in Smith normal form

I have been working through Smith normal form examples and I am wondering if I am finding the correct pivot in order to carry out the calculation. Let $V \subset \mathbb{Z}$ be an Abelian group with ...