Matrix similar to its transpose

The conversation is discussing the similarity of a matrix to its transpose, particularly in relation to a Jordan block. The person speaking has attempted to prove the result but is unsure and is looking for further guidance. In summary, the conversation discusses the similarity of a matrix to its transpose, particularly in relation to a Jordan block, and the person speaking has attempted to prove the result but is unsure and is seeking further guidance.
  • #1
arthurhenry
43
0
Why is every matrix (complex) similar to its transpose?

I am looking at a typical jordan block and I see that the transpose of the nilpotent part is again nilpotent and actually similar to the nilpotent part. I can see that the scalar part of the jordan block does not change under transposing, but I still cannot show the result.

Thank you
 
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  • #2
Did you try to prove it the way micromass suggested the last time you asked? Link.
 
  • #3
Yes, I did try, and at the time I though I solved it. My problem has been after "reducing the problem to a Jordan block"...

Thank you
 

1. What is a matrix similar to its transpose?

A matrix similar to its transpose is a square matrix that has the same elements as its transpose, but in a different order. This means that the rows of the matrix are switched with its columns.

2. How do you determine if a matrix is similar to its transpose?

To determine if a matrix is similar to its transpose, you can check if the matrix is equal to its own transpose. If the matrix is equal to its transpose, then it is considered to be similar.

3. What are the properties of a matrix similar to its transpose?

The properties of a matrix similar to its transpose include having the same determinant, eigenvalues, and trace as its transpose. It also has the property that its diagonal elements are equal to its transpose's diagonal elements.

4. Are all matrices similar to their transpose?

No, not all matrices are similar to their transpose. Only square matrices can be similar to their transpose. Additionally, not all square matrices are similar to their transpose. For example, a matrix with all zeros is not similar to its transpose.

5. How is a matrix similar to its transpose used in real-world applications?

A matrix similar to its transpose is commonly used in statistics, physics, and engineering. It is helpful in solving systems of linear equations and determining the properties of symmetric matrices. It also has applications in image processing and data compression.

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