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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?

  1. 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.
  2. Are there some properties of similar matrices that can characterize similar matrices?

Thanks!

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    @HuiYu They are OK, but they don't allow to characterize similarity completely. I.e. two non-similar matrices can have the same characteristic polynomial.2012-11-27

1 Answers 1

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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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    @JSwanson I think what he meant by "new" is not the result, but his pedagogical approach. E.g. when Sheldon Axler's famous article "Down with Determinants!" was out (published on the same year as Stoustrup's paper), my impression is that many people thought that the way he *taught* linear algebra was new. But certainly the results as well as the theory behind it were well known.2018-05-21