prove $a_{ii}a_{jj}>|a_{ij}|^2$ for a complex positive definite matrix.

Question

Show that if an $n*n$ complex matrix $[a_{ij}]$ is positive definite then $a_{ii}a_{jj}>|a_{ij}|^2$ for all distinct $i$ and $j$ such that $i,j=1,...,n$.

The last part of question was the same for $n=2$. I did that using a determinant argument but I cannot use the same idea in general case.

Answer 1

Note that $M = \pmatrix{a_{ii}&a_{ij}\\a_{ji}& a_{jj}}$ is a principal submatrix of $A$. Thus, if $A$ is positive definite, then so is $M$.