Eigen Vector Proofs: Proving Real Symmetric Matrix M is Positive Definite

[sdevoe]

Homework Statement

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 definite if and only if its eigen-
values are positive.

Homework Equations

Mx=λx

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.

 

[sunjin09]

S is not the eigenvectors of M, S^{-1} is, the rest is straightforward verification.

 

[sdevoe]

With what equation would I begin that proof?

 

[sunjin09]

With what equation would I begin that proof?

first you claim that S^-1 are the eigenvectors, then you prove your claim by definition of eigenvectors

 

[Deveno]

note that M = S^-1DS means that

S^-1D = MS^-1.

express both matrix products above in terms of the column vectors of the matrix on the right in each product. compare your results, what do they say?

 

[sdevoe]

Ok I have that now what about the positive definite aspect?