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Singular values of unitarily equivalent matrices

  1. Jan 20, 2009 #1
    1. The problem statement, all variables and given/known data
    If two square matrices, A and B are unitarily equivalent then A = QBQ* for some unitary Q of the same size as A and B. Prove that A and B are unitarily equivalent if and only if they have the same singular values

    2. Relevant equations

    3. The attempt at a solution
    I started from the definition of singular values:

    Au = sigma v for singular vectors u and v
    A*v = sigma u
    substitution A with QBQ*,
    QBQ*u = sigma v
    (QBQ*)*v = QB*Q*v = sigma u

    Can't see how this leads to proving that the sigmas of A and B are the same..?
  2. jcsd
  3. Jan 20, 2009 #2


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    Science Advisor
    Homework Helper

    If Q is unitary, the Q*=Q^(-1). Operate on the left by Q*.
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