Linear Algebra Done Right · Problem 7C.23

Operators on Inner Product Spaces — Sheldon Axler

Mathematics — Linear Algebra · positive-operators

Problem

For T∈L(V)T \in \mathcal{L}(V) and u,v∈Vu, v \in V, define ⟨u,v⟩T\langle u, v \rangle_T by ⟨u,v⟩T=⟨Tu,v⟩\langle u, v \rangle_T = \langle Tu, v \rangle. (a) Suppose T∈L(V)T \in \mathcal{L}(V). Prove that ⟨⋅,⋅⟩T\langle \cdot, \cdot \rangle_T is an inner product on VV if and only if TT is an invertible positive operator (with respect to the original inner product ⟨⋅,⋅⟩\langle \cdot, \cdot \rangle). (b) Prove that every inner product on VV is of the form ⟨⋅,⋅⟩T\langle \cdot, \cdot \rangle_T for some positive invertible operator T∈L(V)T \in \mathcal{L}(V).

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