TITLE: Prove that a symmetric matrix with a positive diagonal entry has at least one positive eigenvalue
QUESTION [12 upvotes]: Let $A$ be a symmetric martix $n \times n$ such that there is some $i$ such that $a_{ii}>0$.
Prove that $A$ has a positive eigenvalue.

I have a hint which I don't how to use/check: "Check that $a_{ii}=e^t_i*A*e_i$.
Thanks,
Alan

REPLY [13 votes]: By contradiction assume that all the eigenvalues $\lambda_1,\ldots,\lambda_n$ of $A$ are non positive and by spectral theorem let $(v_1,\ldots,v_n)$  an orthonormal basis of eigenvectors then using the hint let $e_i=\alpha_1v_1+\cdots+\alpha_nv_n$ and then
$$a_{ii}=e_i^tAe_i=\sum_{j=1}^n\lambda_j\alpha_j^2\le0$$
which is a  contradiction.