Subject: DIAGONALIZATION OF LAGRANGIAN MATRIX

From: "E. Lewars" <[email-masked]>
Date: Tue, 8 Dec 1998 21:14:31 -0500 (EST)

Tuesday 1998 Dec 8

Hello,

I suspect this question has a simple answer:

 In one derivation of the Hartree-Fock equations the matrix L of Lagrangian
multipliers is written in terms of a diagonal matrix L'

                          L = PL'P**-1

i.e L is diagonalized. This leads to

                          F*psi = e*psi

where psi is a column matrix of the, say, electron 1 wavefunctions
                              psi_1(1)
                              psi_2(1)
                               .
                               .
                               .

   and e (epsilon) is a diagonal matrix
                       (-1/2)l_11
                                  (-1/2)l_22        0
                                              .
                                                .
                               0                  .

(the Lagrangians turn out to be proportional to the energy levels and
books avoid the -1/2 factor, making l_11 = e_1, etc., by  slyly putting in a
-2, I think, ahead of the game).

QUESTION:
             How do we know that the Lagrangian matrix L is diagonalizable?
Is it pretty general that in quantum mechanics square matrices are
diagonalizable (that's not the case in mathematics)?

        Thanks
          E. Lewars

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