Classifying Complex Matrices with Cubed Identity: What Does Similarity Mean?

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In summary, the question is asking to find a set of complex 3x3 matrices that satisfy A^3 = I, where none of the matrices are similar to each other, and any other 3x3 matrix that satisfies A^3 = I is similar to one of the matrices in the set. This is related to finding equivalence classes, and one way to approach this problem is by considering the eigenvalues of A. However, the characteristic polynomial of A cannot simply be (x-1)^3 and there are other complex numbers that satisfy A^3 = I. It is unclear if there is a way to compose matrices from characteristic polynomials.
  • #1
CoachZ
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The question posed is "Classify up to similarity all 3 x 3 complex matrices [tex]A[/tex] s.t. [tex]A^{3} = I[/tex]. I think the biggest problem I'm having is understanding what exactly this is asking me to do. The part that says "Classify up to similarity" is really throwing me off, so if someone could tell me what that implies, it would be very helpful!

Maybe it deals with equivalence classes, or something like that?
 
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  • #2
CoachZ said:
… The part that says "Classify up to similarity" is really throwing me off, so if someone could tell me what that implies, it would be very helpful!

Maybe it deals with equivalence classes, or something like that?

Hi CoachZ! :smile:

Yes, it's putting two matrices in the same equivalence class if they are similar.
 
  • #3
It means, can you find a set of complex 3x3 matrices s.t. A^3 = I, such that none of them are similar to each other, and any other such 3x3 matrix is similar to one of the matrices in your set?

Hint: what can you say about eigenvalues of A?
 
  • #4
hamster143 said:
It means, can you find a set of complex 3x3 matrices s.t. A^3 = I, such that none of them are similar to each other, and any other such 3x3 matrix is similar to one of the matrices in your set?

Hint: what can you say about eigenvalues of A?

I think that I'm a little confused, because if A^3 = I, then the eigenvalues for A would simply be the eigenvalues for I, which is just 1,1,1, since I is a diagonal matrix. Therefore, the characteristic polynomial of such A must be in the form (x-1)^3, right? Is there a way to compose matrices from characteristic polys?
 
  • #5
CoachZ said:
I think that I'm a little confused, because if A^3 = I, then the eigenvalues for A would simply be the eigenvalues for I,

No, the eigenvalues must have cube equal to 1, but there are complex numbers other than 1 itself with that property...

Therefore, the characteristic polynomial of such A must be in the form (x-1)^3, right?

No, for example the characteristic polynomial could be x^3-1 .

Is there a way to compose matrices from characteristic polys?

One would assume the textbook would have such information before asking this question.
 

1. What is "Classify up to similarity"?

"Classify up to similarity" is a method used in data analysis to group data points into categories based on their similarities. This technique is commonly used in machine learning and pattern recognition.

2. How does "Classify up to similarity" work?

This method first compares each data point to a set of pre-defined categories and assigns it to the category that it is most similar to. Then, it looks at the remaining data points and repeats the process until all data points have been categorized.

3. What are the advantages of using "Classify up to similarity"?

One major advantage of this method is that it can handle large and complex datasets. It also does not require prior knowledge of the data or the categories, making it useful for exploratory data analysis.

4. What are the limitations of "Classify up to similarity"?

One limitation is that it relies heavily on the initial set of categories, which may not accurately represent the data. It also may not be suitable for datasets with a high degree of variability or outliers.

5. How is "Classify up to similarity" different from other classification methods?

Unlike other classification methods, "Classify up to similarity" does not require a predefined set of features or attributes. It also considers the overall similarity between data points, rather than focusing on specific features.

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