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we know a square matrix is similar to its transpose, this result holds true over any field. for they have the same "invariant factors".

Similarly, it has been proven that a square matrix is congruent to its transpose too. but the known proof of this result needs highly convoluted math. I wonder whether there is also something invariant under congruence or not. And with this we can give an elementary proof of this result.

for a proof of "A square matrix is congruent to its transpose"

see link text

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why are you not happy with the proof in the link? –  Martin Brandenburg Apr 16 '10 at 12:20
    
I think the proof by R.Horn et al has been very nice and concise. I read the paper many times and also wishes to simplify the proof, but at last, I accept the proof as the best one. –  Sunni Apr 16 '10 at 15:25
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