Is there any way to tell if a matrix is similar to a matrix other than finding an explicit such that ?

The reason is that I am trying to find conjugacy classes of in the easiest possible way.

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Three people were asked whether it was better to have a wife or a mistress.

Doctor: "A wife, because studies have shown that wives have beneficial effects on your health."
Lawyer: "Definitely a mistress, because a wife will just take all your money in a divorce."
Mathematician: "Both. That way, your wife thinks you're with your mistress, your mistress thinks you're with your wife, and there's more time to do math."