Two Theorems Today: [Note: Self Adjoint and Hermitian mean the same thing. <f,g> is the inner product of the vectors f and g.]
Theorem 1: If L is a self adjoint (anti s.a.) operator then the eigenvalues of L are real (imaginary).
Theorem 2: If L is a self adjoint operator then eigenvectors belonging to different eigenvalues are orthogonal.
Proof of 1: Self adjoint means <f,Lg> = <Lf,g> ; To be an eigenvector f associated with an eigenvalue m means Lf = mf.
Start here: <f,Lf> = <f,mf> = m<f,f> ........................(1)
Also <f, Lf> = <Lf,f> [L is self adjoint or Hermitian]
=<mf,f>
=m*<f,f>.....................................(2)
From (1) and (2) we conclude that m* = m. So m is real. The Anti-Hermitian case is left as an exercise.
Proof of 2: Say Lf=mf and Lg=ng where f and g are eigenvectors for the two different eigenvalues m and n. L is Hermitian so m and n are real. (i.e. m* = m and n* = n.).
<f,Lg> = <f,ng> = n<f,g>............................................(1)
<f,Lg> = <Lf,g> = <mf,g> = m*<f,g> = m<f,g>........(2)
from (1) and (2) we get n<f,g> = m<f,g> and so (n - m)<f,g> = 0 and since m and n are different <f,g> = 0. QED.
11:33:13 AM 
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