1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Prove: similar matrices have the same characteristic polynomial

  1. Jul 10, 2006 #1
    Prove: Similar matrices have the same characteristic polynomial.

    By characteristic polynomial of A i mean det(A-tI) where t is a scalar.
    A is similar to B if A = Q^-1 B Q for some invertible matrix Q. (i.e. B is the matrix representation of the same linear transformation as A but under a different basis.)

    I do know that similar matrices have the same determinant. This can be easily proved using det(AB) = det(A)det(B). But when you change the diagonal entries of the determinant im not sure how it will be affected...
  2. jcsd
  3. Jul 10, 2006 #2

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    You've just said you know the answer. (A and B similar implies they have the same char poly, which is what you were asked to prove.)
  4. Jul 10, 2006 #3


    User Avatar
    Science Advisor

    det(AB)= det(A)det(B) so [itex]det(Q^{-1}(1-\lambda P Q}=det(Q^{-1})det(1- \lambda P) det(Q)[/itex].
  5. Jul 11, 2006 #4
    thanks i got it now (after a few manipulations of my own).
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook