How to prove that elements in the main diagonal of PD matrix are all positive?

Question

Reminder: As PD matrices are defined vector X is not the vector 0

Given a positive definite matrix A which is symmetric We need to prove that he following elements in the main diagonal are all positive

(A(1,1) A(2,2) ... A(n,n))

I started solving this but got stuck at the end, any help?

Answer 1

$A=[a_{i,j}]$ is the matrix of the scalar product $<x,y>=x^TAy$.

Then $<e_i,e_i>=||e_i||^2=e_i^TAe_i=a_{i,i}>0$.