Calculation and Uniqueness of Smith Normal Forms

In summary, the conversation discusses finding the Smith Normal Form of a given matrix over the ring of integers. The concept of Smith Normal Forms being unique up to multiplication by a unit in R is also mentioned. The individual has performed elementary row and column operations to obtain a Smith Normal Form, but it does not match the given answer. They are unsure if their method is correct and are seeking clarification on calculating Smith Normal Forms.
  • #1
jj4
2
0
FYI this is a homework problem which I already have the answer to but would like to clarify some points on.

Homework Statement



Find the Smith Normal Form of the matrix

[tex]
\left[ \begin{array}{cccc} 6 & 0 & 4 \\ 0 & 6 & 8 \\ 0 & 3 & 0 \end{array} \right]
[/tex]

over the ring of integers.

Homework Equations



As I understand it Smith Normal Forms are unique up to multiplication by a unit in R.

The Attempt at a Solution



I have performed a series of elementary row and column operations to get to the following Smith Normal Form:

[tex]
\left[ \begin{array}{cccc} 6 & 0 & 0 \\ 0 & 6 & 0 \\ 0 & 0 & -24 \end{array} \right]
[/tex]

However the "answer" given is actually:

[tex]
\left[ \begin{array}{cccc} 1 & 0 & 0 \\ 0 & 6 & 0 \\ 0 & 0 & -24 \end{array} \right]
[/tex]

The only units in the ring are 1 and -1 so the answers are not equivalent and presumably I have gone wrong somewhere. Can someone help me out here. My problem is not only related to this example, I have the same issue with others. Is it because my Smith Normal Form and the original matrix are not actually equivalent?
 
Physics news on Phys.org
  • #2
I'm just looking for a basic explanation of how to calculate Smith Normal Forms. Is this thread in the correct sub-forum?
 

1. What is a Smith Normal Form?

A Smith Normal Form (SNF) is a special form of a matrix that is unique to each matrix. It is a diagonal matrix with non-negative integers along the diagonal and zeros in all other positions. It is also the equivalent of the reduced row echelon form for integer matrices.

2. How is the Smith Normal Form calculated?

The calculation of the Smith Normal Form involves a series of elementary row and column operations on the original matrix. These operations include swapping rows and columns, multiplying rows and columns by constants, and adding multiples of one row or column to another. The end result is a diagonal matrix in SNF.

3. Why is the Smith Normal Form important?

The Smith Normal Form is important because it provides a unique representation of a matrix and is useful in solving systems of linear equations. It also has applications in algebraic number theory, coding theory, and cryptography.

4. Can every matrix be transformed into a Smith Normal Form?

Yes, every integer matrix can be transformed into a Smith Normal Form through a series of elementary operations. However, the resulting SNF may not be unique if the matrix has zero rows or columns.

5. How is the uniqueness of the Smith Normal Form proven?

The uniqueness of the Smith Normal Form can be proven using the fundamental theorem of finitely generated abelian groups. This theorem states that every finitely generated abelian group can be expressed as a direct sum of cyclic groups. The SNF is essentially a representation of the abelian group structure of the original matrix, making its uniqueness provable by this theorem.

Similar threads

  • Calculus and Beyond Homework Help
Replies
6
Views
474
Replies
1
Views
688
  • Calculus and Beyond Homework Help
Replies
7
Views
1K
  • Calculus and Beyond Homework Help
Replies
1
Views
1K
  • Calculus and Beyond Homework Help
Replies
5
Views
896
  • Calculus and Beyond Homework Help
Replies
8
Views
119
  • Calculus and Beyond Homework Help
Replies
0
Views
43
Replies
6
Views
1K
  • Calculus and Beyond Homework Help
Replies
7
Views
710
  • Linear and Abstract Algebra
Replies
4
Views
2K
Back
Top