Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Quick question about Hermitian operators

  1. Aug 14, 2012 #1
    If H is a Hermitian operator, then its eigenvalues are real. Is the converse true?
     
  2. jcsd
  3. Aug 14, 2012 #2
    No. Take as counterexample the matrix [tex]\left(\begin{array}{cc} 0 & 1\\ 0 & 0 \end{array}\right).[/tex]
    However, if you know that the operator is normal (that is: if [itex]AA^*=A^*A[/itex]), then it is true that it is Hermitian iff the eigenvalues are real.
     
  4. Aug 17, 2012 #3
    Good counterexample!

    A few more useful rules: a matrix ##M## has "good" eigenvectors if and only if it is normal. "Good" eigenvectors means it is possible to choose eigenvectors such that they form an orthonormal basis for whatever vector space ##M## acts on.

    Equivalently, a matrix is normal if and only if it can be diagonalized by a unitary transformation:
    ##
    M = U D U^{-1}
    ##
    for some unitary matrix ##U## and diagonal matrix ##D## whose diagonal elements are eigenvalues of ##M##.

    Hermitian matrices are always normal, and their eigenvalues are always real. But if ##M## is abnormal, it might have real eigenvalues and "bad" eigenvectors which are not orthogonal. Here's an example:

    ##
    \left[\begin{array}{cc}
    1 & 1 \\
    0 & 2 \\
    \end{array}\right]
    ##

    This thing has eigenvalues 1 and 2, but its eigenvectors

    ##
    \left[\begin{array}{c}
    1 \\ 0 \\
    \end{array}\right]
    \quad
    \left[\begin{array}{c}
    1 \\ 1 \\
    \end{array}\right]
    ##

    are not orthogonal to each other.

    The example micromass gave is even more badly-behaved: it is defective, which means it doesn't have enough linearly-independent eigenvectors to span the vector space. So we can't even form a "bad" basis using its eigenvectors.
     
    Last edited: Aug 17, 2012
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook