(adsbygoogle = window.adsbygoogle || []).push({}); 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.

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# Homework Help: Eigen vector proofs

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