I'm studying for my exam of linear algebra.. I want to prove the following corollary:
If $A$ is a symmetric positive definite matrix then each entry $a_{ii}> 0$, ie all the elements of the diagonal of the matrix are positive.
My teacher gave a suggestion to consider the unit vector "$e_i$", but I see that is using it.
$a_{ii} >0$ for each $i = 1, 2, \ldots, n$. For any $i$, define $x = (x_j)$ by $x_i =1$ and by $x_j =0$, if $j\neq i$, since $x \neq 0$, then:
$0< x^TAx = a_{ii}$
But my teacher says my proof is ambiguous. How I can use the unit vector $e_1$ for the demonstration?