Is Every Matrix Similar to Its Transpose?

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Discussion Overview

The discussion revolves around whether every matrix is similar to its transpose, referencing a claim found in Wikipedia. The scope includes theoretical considerations related to linear algebra and matrix theory.

Discussion Character

  • Debate/contested, Technical explanation

Main Points Raised

  • One participant questions the claim, suggesting that not every matrix is similar to its transpose and implying that there may be specific types of matrices for which this holds true.
  • Another participant argues that the base field being algebraically closed is not a limiting factor, stating that similarity can be analyzed over larger fields and that the question can be reduced to an algebraically closed field.
  • This participant also presents a technical argument that two matrices are similar if their Jordan normal forms have the same invariant factors, asserting that a matrix and its transpose share the same invariant factors, thus concluding that a matrix is similar to its transpose.
  • A later reply expresses gratitude for the reference provided, indicating that the source was helpful.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the claim. There are competing views regarding the similarity of matrices to their transposes, with some asserting it is true under certain conditions while others suggest it is not universally applicable.

Contextual Notes

The discussion does not resolve the assumptions regarding the types of matrices or the implications of the field being algebraically closed. The argument about invariant factors and Jordan normal forms is presented without detailed mathematical steps.

arthurhenry
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Is it true that every matrix is similar to its transpose? A claim in Wikipedia...


(field is alg. closed)
 
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Deleted. Funny that I made mistake I've made before.
 
Last edited:
Not every matrix. Only specific types.
I suspect there is a context mentioned?
 
Whether the base field is algebraically closed doesn't matter here, since two matrices are similar over a small field if and only if they are similar over a bigger field. So the question can always be reduced to an algebraically closed field.

That said, it can be shown that two matrices are similar if and only if the Jordan normal form has the same "invariant factors" and it can be shown that a matrix and it's transpose have the same invariant factors. So a matrix is similar to it's transpose.

Check out the book "matrix analysis" by Horn and Johnson.
 
I thank you Micromass, that source was very helpful.
 

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