Why inner product on R^n have uniform prototype with symmetric matrix A and positive eigenvalues?

Question

Details of the problems are given below. Assume A is a n*n symmetric matrix. Show that any inner product on R^n has this formula for some symmetric matrix A with all positive eigenvalues. The formula acts like (x,y)=y$^{\rm T}$Ax, where x and y are vectors in R^n.

Answer 1

This is for linear algebra at U of M, you shouldn't post questions on here, just ask at office hours. I'll give you a hint, though: Consider an arbitrary inner product on $\mathbb{R}^{n}$ and then use the axioms of an inner product and another special property (all inner products on $\mathbb{R}^{n}$ are bilinear, meaning they are linear in both arguments of the inner product) to prove the desired result. Also, you might want to consider the standard basis for $\mathbb{R}^{n}$, because the entries of the matrix $A$ involve this basis. Good luck!