Linear Algebra | Part IB, 2004

(i) Let VV be an nn-dimensional inner-product space over C\mathbb{C} and let α:VV\alpha: V \rightarrow V be a Hermitian linear map. Prove that VV has an orthonormal basis consisting of eigenvectors of α\alpha.

(ii) Let β:VV\beta: V \rightarrow V be another Hermitian map. Prove that αβ\alpha \beta is Hermitian if and only if αβ=βα\alpha \beta=\beta \alpha.

(iii) A Hermitian map α\alpha is positive-definite if αv,v>0\langle\alpha v, v\rangle>0 for every non-zero vector vv. If α\alpha is a positive-definite Hermitian map, prove that there is a unique positivedefinite Hermitian map β\beta such that β2=α\beta^{2}=\alpha.

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