1. The problem statement, all variables and given/known data Let M be a symmetric matrix. The eigenvalues of M are real and further M can be diagonalized using an orthogonal matrix S; that is M can be written as M = S^-1*D*S where D is a diagonal matrix. (a) Prove that the diagonal elements of D are the eigenvalues of M. (b) Prove that the real symmetric matrix M is positive definite if and only if its eigen- values are positive. 2. Relevant equations Mx=λx 3. The attempt at a solution a) So i know that S is the eigen vectors of M and D is the eigen values but I do not know how to prove that. b)Does this have something to do with the gradient. I know positive definite means that transpose(x)*M*x must be greater than zero where x is x(1) through x(n) and M is the matrix in question.