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So obviously "=>" is always true, but I can't find any information on "<=". It seems logical, but I can't find a way to prove it. Also, even finding a counter-example doesn't seem easy, because then you'd have to prove there isn't any invertible matrix P so that [tex]P^{-1} A P = B[/tex], so I suppose a counter-example should be done with reductio ad absurdum, but nothing strikes me as an obvious example.

Any help?

Thank you,

mr. vodka