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Unitary equivalence vs. similarity

  1. Apr 23, 2007 #1
    Two matrices A and B are defined to be unitarily equivalent if there exists a unitary matrix P such that A = P*BP.

    If one is given two matrices A and B with no P, how do we know if the matrices are unitarily equivalent? I guess what I'm asking is, is it sufficient to check whether the matrices have the same characteristic polynomial and that one of them is diagonal? What exactely does normality have to do with unitary equivalence? How is it related to similarity and congruence of matrices? Someone suggested to me that I check to see if they are both normal, but I guess I don't see how that fits into the definition. I know that matrices that are unitarily equivalent are also similar (P seems a lot like regular change of basis matrix Q except that it comes from an orthonormal basis instead of regular bases).

    Anyone know anything I'm missing?
  2. jcsd
  3. Apr 23, 2007 #2
    I do not think that if one of the given matrices is diagonal then just comparing the two matrices to have the same char. function is enough to gaurantee that the matrices are unitarily equivalent. This may happen if A is not normal.

    In general if none of the matrices are diagonal then simply comparing the char function will only give us similarilty and not unitary equivalence. If in addition the matrices are normal (unitarily diagonalizable) then we can say they are unitarily equivalent.

    If A and B are normal and have same e.v. : A=P* D P and B=Q* D Q (P, Q unitary)
    then P A P* = D and then B = Q* P A P* Q. Now P*Q is unitary. So A and B are unitarily equivalent.

    But this seems to be a sufficient condition only. May be one can show that this is necessary as well.
  4. Apr 25, 2007 #3
    aha. this is making much more sense now. It seems that both A and B must be normal; to further check, one needs only check whether they have the same eigenvalues, since if they are unitarily equivalent then they are certainly similar. :rofl:
  5. Apr 26, 2007 #4
    Just wanted to straighten things out for record. I said earlier that if char. polynomial is same then the two matrices are similar. But that is not true. The implication is only that if two matrices are similar then they have the same char. polynomial. Again if in addition we are given the matrices are diagonalizable (not necessarily normal) then we have similarity.

    So for now we have:

    If Diagonalizable: identical eigenvalues <=> Similarity
    If Normal: identical eigenvalues <=> Unitary equivalence
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