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Let (V,<,>) be a complex inner product space with an orthonormal basis {v1,v2,.......vn}. Let L:V------>V be a linear operator. Explain what is meant by saying that L is self-adjoint. Let A=a_ij be the matrix representing L with respect to the basis {v1,......v_n}. prove the following.

i) L is self-adjoint if and only if <Lvi,vj>=<vi,Lvj>.

ii) a_ij=<Lvj,vi>.

L is self adjoint means L = L*, but we know L* is the one and unique operator for which <Lv, u> = <v, L*u> for all u,v. How do i prove i) and ii).

Thanks

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# Linear Operator and Self Adjoint

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