How to tell if two matrices are similar? [duplicate]

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Two n-by-n matrices A and B are called similar if $$ \! B = P^{-1} A P $$ for some invertible n-by-n matrix P.

Similar matrices share many properties:

Rank

Determinant

Trace

Eigenvalues (though the eigenvectors will in general be different)

Characteristic polynomial

Minimal polynomial (among the other similarity invariants in the Smith normal form)

Elementary divisors

Given two square matrices A and B, how would you tell if they are similar?

Constructing a $P$ in the definition seems difficult even if we know they are similar, does it? Not to mention, use this way to tell if they are similar.
Are there some properties of similar matrices that can characterize similar matrices?

Thanks!

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1 Answer

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For a pedagogical discussion, see Jakob Stoustrup, Linear Algebra in the Classroom: A Note on Similarity. (The author claims on his website that this paper appeared in International Journal of Mathematical Education in Science and Technology, 26(6): 917-920, 1995, but it wasn't found on the journal's website.)

The author essentially proposes a generalised eigenspace approach (see lemma 1), so that no matrix transformation is needed.

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