A question about invariant factors

In summary, the conversation discusses a theorem in a textbook that states a finitely generated torsion R-module can be written as a direct sum of cyclic modules. The example given in the textbook illustrates how to find the invariant factors using elementary divisors. However, if the elementary divisors are not given, a presentation matrix can be used to diagonalize and find the invariant factors. Examples of this can be found on the speaker's website.
  • #1
Artusartos
247
0
A Theorem in our textbook says...

If R is a PID, then every finitely generated torision R-module M is a direct sum of cyclic modules

[tex]M= R/(c_1) \bigoplus R/(c_2) \bigoplus ... \bigoplus R/(c_t)[/tex]

where [tex]t \geq 1[/tex] and [tex]c_1 | c_2 | ... | c_t [/tex].

There is an example from our textbook that I attached...they find the invariant factors from the elementary divisors. But what if we had to find the invariant factors without being given the elementary divisors. How would we do that?

Thanks in advance
 

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  • #2
well you have to be given something. i have some examples in my book on my webpage.

to be "given" a f.g. module usually means to be given a "presentation" as a quotient of two free modules.

such a quotient is specified by a matrix. then you diagonalize that presentation matrix.

see the discussion here:

http://www.math.uga.edu/%7Eroy/845-1.pdf
 
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1. What are invariant factors?

Invariant factors refer to the unique factors of a matrix that remain the same regardless of any similarity transformations applied to the matrix.

2. Why are invariant factors important?

Invariant factors are important because they provide essential information about the structure and properties of a matrix. They can be used to determine the rank, eigenvalues, and other characteristics of a matrix.

3. How are invariant factors calculated?

The calculation of invariant factors involves finding the elementary divisors of a matrix, which are the unique factors of the matrix that cannot be further factored. These elementary divisors are then arranged in a descending order to form the invariant factors.

4. Can invariant factors be used for any type of matrix?

Yes, invariant factors can be used for any type of matrix, including square matrices, rectangular matrices, and even complex matrices. However, the method of calculation may vary depending on the type of matrix.

5. What is the significance of invariant factors in linear algebra?

Invariant factors play a crucial role in linear algebra as they provide a concise representation of a matrix's structure and properties. They are useful in solving systems of linear equations, computing determinants, and understanding the behavior of linear transformations.

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