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: Unitary, invariant subspaces

  1. May 7, 2007 #1
    1. The problem statement, all variables and given/known data

    Let U be a unitary operator on an inner product space V, and let W be a finite-dimensional U-invariant subspace of V. Prove that

    (a) U(W) = W
    (b) the orthogonal complement of W is U-invariant
    (for ease of writing let the orthogonal complement of W be represented by W^.

    2. Relevant equations

    Unitary: U*U = UU* = I

    3. The attempt at a solution

    (a) first show that U(W) is contained in W, and then show that W in contained in U(W).
    - U(W) is contained in W because W is U-invariant
    - show W is contained in U(W)
    choose x in W and show it is contained in U(W)
    U(x) is in W
    Is this circular thinking?
    Should I instead show that the range of U (restricted to W) is W itself? Or that the nullity of U (restricted to W) is 0?

    Note: U restricted to W (let’s call it U_w) is also unitary.

    W^ = {x in V : <x, y>=0 for all y in W}

    Now show that U(W^) is contained in W^

    I’m not sure what to do now.

    Thanks for your help!
  2. jcsd
  3. May 7, 2007 #2


    User Avatar
    Science Advisor
    Homework Helper

    Do the last one.
    Use the definitions. You want to say that if x is in W^, then so is U(x). Well x is in W^ iff for all y in W, <x,y> = 0, and U(x) is in W^ iff for all y' in W, <U(x),y'> = 0.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook