Is Every Matrix Similar to Its Transpose?

arthurhenry · Jun 20, 2011

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

(field is alg. closed)

Fredrik · Jun 20, 2011

Deleted. Funny that I made mistake I've made before.

I like Serena · Jun 20, 2011

Not every matrix. Only specific types.
I suspect there is a context mentioned?

micromass · Jun 20, 2011

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.

arthurhenry · Jun 20, 2011

I thank you Micromass, that source was very helpful.

Is Every Matrix Similar to Its Transpose?

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