Proof that every Eigenvalue of self adjoint operator is real
Published On :2021-06-26 23:41:00
Theorem: Every Eigenvalue of self adjoint operator is real
[begin{array}{l}\
Suppose;T;is;self;adjoint Rightarrow T = {T^ * }\\
;Let;lambda in F;;;be;eigenvalue;of;T\\
Rightarrow exists ;v in ;{cal V};s.t;;;Tv = lambda v\\
note;that:lambda {left| v right|^2} = lambda leftlangle v right.,left. v rightrangle \\
= ;langle lambda v,vrangle \\
= ;langle Tv,vrangle \\
= ;langle v,{T^ * }vrangle \\
= ;langle v,Tvrangle \\
= ;langle v,lambda vrangle \\
= ;bar lambda langle v,vrangle = ;bar lambda {v^2}\\
lambda = ;bar lambda to lambda ;is;real\
end{array}]