Find the elementary divisors and invariant factors

• MHB
• fabiancillo
In summary, finding the elementary divisors and invariant factors of a matrix is important in understanding its properties and behavior. To find the elementary divisors, the eigenvalues of the matrix are first found and used to construct the elementary divisors. These elementary divisors are the diagonal entries of the Jordan canonical form of the matrix. The invariant factors are the product of these elementary divisors and provide more information about the structure of the matrix. Different sets of elementary divisors and invariant factors are possible for the same matrix, but they will always have the same product. In real-world applications, invariant factors have various uses in fields such as engineering, physics, and economics, including analyzing systems of linear differential equations, modeling physical systems, and solving optimization problems
fabiancillo
Hello I have problems with this exercise

Find the elementary divisors and invariant factors of each of the following groups

a) $G1= Z_6 \times Z_{12} \times Z_{18}$ , b) $G_2= Z_{10} \times Z_{20} \times Z_{30} \times Z_{40}$Thanks

a) I think that elementary divisors are $\{2,3,2^2,3,2,3^2 \}$ because is the prime decomposition of ${6,12,18}$.

b) elementary divisors are $\{5 ,2 ,2^2, 5, 2, 3, 5, 2^3, 5 \}$
. But I don't use the Chinese remainder theorem to split each factor into cyclic pp-groups, then regroup

Last edited:
You are correct so far in parts (a) and (b). You just need to find the invariant factors now.

1. What are elementary divisors and invariant factors?

Elementary divisors are the prime factors of a matrix, while invariant factors are the products of these prime factors. They are used to describe the structure and properties of a matrix.

2. How do you find the elementary divisors and invariant factors of a matrix?

To find the elementary divisors and invariant factors, you can use the Smith normal form algorithm. This involves performing elementary row and column operations on the matrix until it is in a diagonal form with the elementary divisors on the diagonal.

3. What is the significance of finding the elementary divisors and invariant factors of a matrix?

Knowing the elementary divisors and invariant factors of a matrix can help in understanding its properties and behavior. It can also be used in solving systems of linear equations and in finding the eigenvalues and eigenvectors of a matrix.

4. Can the elementary divisors and invariant factors of a matrix change?

Yes, the elementary divisors and invariant factors of a matrix can change if the matrix is modified through row and column operations. However, the number of elementary divisors and invariant factors will remain the same.

5. Are there any real-world applications of finding the elementary divisors and invariant factors of a matrix?

Yes, finding the elementary divisors and invariant factors of a matrix has applications in fields such as engineering, physics, and economics. It is used in solving systems of linear equations, analyzing networks and circuits, and studying the stability of systems.

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