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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!

  • 3
    You can have a look here: [Frobenius noral form](http://en.wikipedia.org/wiki/Frobenius_normal_form).2012-11-27
  • 0
    @DanShved: Thanks! So decomposing each matrix into some canonical form, such as Jordan form, is a way, by definition.2012-11-27
  • 0
    I think for finite dimensional spaces characteristic polynomials are the best.2012-11-27
  • 4
    @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

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