- #1

blanik

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Suppose V is a 3-dimensional complex inner-product space. Let B1 = {|v1>,|v2>,|v3>} be an orthonormal basis for V.

Let H1 and H2 be self-adjoint operators represented in the basis B1 by the Hermitian matrices.

I won't list them, but they are 3x3 matrices. I have found the eigenvalues and eigenvectors of each. They both have degenerate eigenvalues. H1 has Lambda = -5,-5,15 and H2 has Lambda = 10,10,20.

My questions are:

(1) By considering the commutator, show that the two matrices can be simultaneously diagonalized. I know that by definition of the commutator, [H1,H2] = (H1)(H2) - (H2)(H1) = 0. Also, I know that D = (U-1)H(U), where D is the Diagonalized matrix. I need a hint on how to begin to show that they are simultaneously diagonalizable.

(2) As I mentioned before, I have found the eigenvectors for each. I am asked to find the eigenvectors common to both matrices and determine a set of three orthogonal eigenvectors. The eigenvectors I found are not the same for both matrices, but they are very similar in that they are basically flipped with a negative sign in one value. Do I need to do something to make these eigenvectors the same for both matrices or did I possibly make a math error.

(3) My last question is how do I verify under a unitary transformation to this basis that both matrices are diagonalized.

Any formulas, procedures, definitions, etc... would be greatly appreciated.