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!

Trace of a linear operator

  1. Jun 19, 2007 #1
    I understand the definition of trace and linear operator individually but I don't seem to understand as to what does it mean by trace of a linear operator on a finite dimensional linear space.
    What I have found out is that trace of a linear operator on a finite dimensional linear space is the trace of any matrix which represents the operator relative to an ordered basis of the space. I am confused as why is this definition well defined.
    If T:V->V is the linear operator defined on V by T(A)=BA for all A in V and B is a fixed matrix. How do I represent T relative to standard ordered basis for V where V is the linear space of all 2X2 real matrices.
     
  2. jcsd
  3. Jun 19, 2007 #2

    CompuChip

    User Avatar
    Science Advisor
    Homework Helper

    As for the first part: you can prove it as follows: Let L be a linear operator and A its matrix representation w.r.t. some chosen basis.
    - Check that the representation w.r.t. any other basis can be written as [tex]D A D^{-1}[/tex], where D is an invertible ("change of basis") matrix.
    - Now check the cyclic property for the trace (e.g. by writing out in components) [tex]\mathrm{Tr}(A B C) = \mathrm{Tr}(B C A) = \mathrm{Tr}(C A B)[/tex]
    - Combine them, you'll see that the trace is the same in any basis. So it's well-defined.

    Now in general, you can write out the matrix of a linear transformation by finding out how it acts on the basis vectors. For example, suppose A mirrors the plane in the origin. Take the standard basis i = (1, 0), j = (0, 1). Now Ai = (-1, 0) and Aj = (0, -1). Putting these as columns of A gives [tex] A =\left(\begin{array}{cc} -1 & 0 \\ 0 & -1 \end{array}\right)[/tex]. Now check for yourself, that this does indeed produce the correct result for any vector (hint: write it out in components w.r.t. to the basis {i, j}).

    Hope that gets you started. I left the details out on purpose, if you get stuck anywhere just ask :)
     
    Last edited: Jun 19, 2007
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Trace of a linear operator
  1. Trace operator (Replies: 1)

  2. Linear operator (Replies: 4)

  3. Linear operator (Replies: 6)

Loading...