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: 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


    User Avatar
    Science Advisor
    Homework Helper

    If Q is unitary, the Q*=Q^(-1). Operate on the left by Q*.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook