A complex square matrix A is called hermitian matrix iff A=A∗ .
Where A∗= Transpose conjugate of the square matrix A .
Let λ be a eigen value of A and X be corresponding eigen vector of λ .
Then AX=λX ...............(1).
Premultiplying both side of (1) by X∗ ,we get
X∗AX=λX∗X.................(2)
Taking conjugate transpose of both side of (2) ,we get
(X∗AX)∗=(λX∗X)∗
⟹X∗A∗(X∗)∗=λˉX∗(X∗)∗
⟹X∗AX=λˉX∗X
(∵(X∗)∗=X and A∗=A)
From equation (2) and (3) we have
λX∗X=λˉX∗X
⟹(λ−λˉ)X∗X=O
But X is not a zero vector as it is eigen vector ,therefore X∗X=O⟹λ−λˉ=0 so that ,λ=λˉ and consequently λ is real.
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