Is Every Complex Matrix Similar to Its Transpose?

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SUMMARY

Every complex matrix is similar to its transpose due to the properties of Jordan blocks. In a Jordan block, the nilpotent part remains nilpotent upon transposition and is similar to the original nilpotent part. The scalar component of the Jordan block is invariant under transposition. The discussion highlights the challenge of proving this similarity, particularly when reducing the problem to a Jordan block structure.

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arthurhenry
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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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Did you try to prove it the way micromass suggested the last time you asked? Link.
 
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
 

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