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The Eigenvectors of a Self-Adjoint (Hermitian)

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Let T >= 0 be a strictly positive definite linear operator on a finite dimensional inner product space V over F = R or C.

(a) Prove that the exponential map Exp: A -> e^A = sum infinity k = 0 1/kl A^k is one-to-one from the space of self-adjoint operators H = {T : T* = T} into the set of positive definite operators

P = {T : T* = T and (Tx,x) >= 0 for all x E V}

In particular e^A is self-adjoint if A is self-adjoint.

(b) Prove that the exponential map Exp is a surjective map from H to P.

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In this solution eigenvectors of a self-adjoint (Hermitian) operator in a finite dimensional space are assessed. The response is enclosed within an attached pdf. file.

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(a) The eigenvectors of a self-adjoint (Hermitian) operator in a finite dimensional ...

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