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Similar matrix and characteristic polynomial

  1. Dec 4, 2007 #1
    A true statement: Two similar matrices have the same characteristic polynomial.

    The converse however is not true in general: two matrices with the same characteristic polynomial need not be similar.

    HOw can I prove this?

    Any help appreciated.
  2. jcsd
  3. Dec 4, 2007 #2
    it's enough to find two matrices which are not similar yet have the same char poly
  4. Dec 4, 2007 #3


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    Note: Pick an easy charisteristic polynomial to aim for.
  5. Dec 5, 2007 #4
    can you suggest an easy characteristic polynomial, i'm stumped on this concept. thank you
  6. Dec 5, 2007 #5
    You can try a difference of squares. For instance, x^2-1 Also, you can think of what you can do to a matrix that wouldn't change the characteristic polynomial.
  7. Dec 5, 2007 #6


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    If two matrices have the same characteristice polynomial, the obviously they have the same eigenvalues. What you want is that they have different eigenvectors. You can do that is one is diagonalizable but the other isn't. To give an example in 2 by 2 matrices, they must have only a single eigenvalue. On matrix would then have 2 \independent eigenvectors, the other only 1 eigenvalue.
  8. Dec 6, 2007 #7
    to prove the part that is true, use change of basis. matrices A and B are similar if there exists an invertible Q such that B=Q^-1*A*Q. But all you're really doing is putting your linear transformation into another basis, so they'll have the same characteristic polynomial. You can check this by showing the change of basis, then taking the determinant of A and B.
  9. Dec 10, 2007 #8
    Take a 2x2 jordan block and also a 2x2 identity matrix, and...
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