1. Not finding help here? Sign up for a free 30min 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!

Unitary Operator

  1. Mar 31, 2012 #1
    Here's the definition.
    Let T be a linear operator on a finite-dimensional inner product space V. If [itex] \|\vec{T(x)}\| = \|\vec{x}\| \\[/itex] for all x in V, we call T a unitary operator.
    The question is asking about for all x in some orthornormal basis for V. Isn't that the same as for all x in V?
     
  2. jcsd
  3. Mar 31, 2012 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Not at all. It says "some orthonormal basis for V". It doesn't say "for all orthonormal bases for V". Think counterexample.
     
  4. Apr 1, 2012 #3
    Doesn't a basis generate all of V, though?
     
  5. Apr 1, 2012 #4

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Yes, a basis generates V. But ||U(a)||=||a|| and ||U(b)||=||b|| doesn't imply that ||U(a+b)||=||a+b||.
     
  6. Apr 1, 2012 #5
    I think the question is confusing me. I interpret "for all x in some orthornormal basis for V" as meaning for every linear combination x of the basis.

    What kind of counterexample am I looking for? Just some orthonormal basis that's not unitary?
     
  7. Apr 1, 2012 #6

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Pick ANY orthonormal basis. Call it {e_1,e_2,...,e_n}. You want to define U somehow. The only condition that U has to satisfy is that ||U(e_i)||=1 for 1<=i<=n.
     
    Last edited: Apr 1, 2012
  8. Apr 1, 2012 #7
    How about I take the standard basis for R2 and define

    T(x1, x2) = (x1 - x2, 0).

    T(1,0) = (1,0); ||T(e1)||= 1
    T(0,1) = (-1,0); ||T(e2)||= 1

    T(1,1) = (0,0); ||T(e1 + e2)||= 0
     
  9. Apr 1, 2012 #8

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Looks good to me!
     
  10. Apr 1, 2012 #9
    Thanks!

    I have another question. I didn't understand property #5. Why does it sum over every entry, not just the diagonal?

    http://planetmath.org/encyclopedia/TraceOfAMatrix.html [Broken]
     
    Last edited by a moderator: May 5, 2017
  11. Apr 1, 2012 #10

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Use sigma notation to write out the product AA*, then take the trace. It just works out that way.
     
    Last edited by a moderator: May 5, 2017
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Unitary Operator
  1. Unitary operators (Replies: 12)

  2. Unitary operator (Replies: 1)

  3. Unitary operators (Replies: 1)

Loading...