Real Vector Spaces and the Real Spectral Theorem

[jesusfreak324]

Homework Statement

Proof:Suppose that V is a real inner product space and T$$\in \wp$$(V). If (v₁... v_n) is a basis for V consisting of eigenvectors for T, then there exists an inner product for V such that T is self-adjoint.

Homework Equations

The Real Spectral Theorem: Suppose that V is a real-inner product space and T $$\in \wp$$(V). Then V has an orthonormal basis consisting of eigenvectors of T if.f. T is self-adjoint.

Eigenvalue / Eigenvector Problem: T(v) = $$\lambda$$v

The Attempt at a Solution

Ummm... I have spent days on trying to figure this out, and the only advice my professor gave to me was to use the real spectral theorem. But the only way to do this is by, as my professor goes on, to come up with an inner product that "turns" the basis list into an orthonormal one. But this seems like a complete contradiction... idk ~~~~

 

[arkajad]

I guess your n is the same as the dimension of V? Suppose it is so. Let [itex](v,w)[/tex] be the original scalar product. Then you define new scalar product of two vectors v,w by<br /> <br /> $$\langle v,w\rangle=\sum_{ij}\delta_{ij} (v,v_i)(w,v_j)$$<br /> <br /> Your basis is now orthonormal - check it. Then play with it.[/itex]