Properties of Hermitian and Positive Definite matrix Let $A \in \mathbb{C}^{nxn}$ be Hermitian and positive definite. I have to show that


*

*$|a_{jk}|^2 < a_{jj}a_{kk}$

*$max_{i,j=1,\dots,n}|a_{ij}| = a_{kk}$ for some $k$ with $1\leq k\leq n$ 


For the first part, since A has entries in $\mathbb{C}$ then $|a_{jk}|^2 = \overline{a_{jk}}a_{jk}$ 
I also know that all of the diagonal values are real and positive, so $a_{jj}>0$ $\forall$  $1 \leq j \leq n $ but im having trouble proving that the multiple of two diagonal values if necessarily larger than the square of the modulus of a non diagonal value. I think it's related to diagonal dominance but im not too sure..
 A: Since $A$ is Hermitian and positive definite, then it admits a unique Cholesky decomposition:
$$A = LL^*$$
where $L$ is full-rank, lower triangular and all the elements on the diagonal are strictly positive. Let $[L]_i$ be the $i$-th row of $L$. Consider also the inner product $(v,w) = \sum_i v_iw_i^*$
Then:
$$\left\{
\begin{array}{l}
([L]_k, [L]_k) = a_{kk}\\
([L]_j, [L]_j) = a_{jj}\\
([L]_j, [L]_k) = a_{jk}\\
\end{array}
\right..$$
Using the Cauchy-Schwarz inequality, one gets that:
$$|([L]_j, [L]_k)|^2 \leq ([L]_j, [L]_j) \cdot ([L]_k, [L]_k) \Rightarrow$$
$$|a_{jk}|^2 \leq a_{jj} \cdot a_{kk}$$
Notice that "$=$" holds only if $[L]_j$ and $[L]_k$ are linearly dependent. Since $L$ is full rank, this can happen only for $j=k$, and hence:
$$|a_{jk}|^2 < a_{jj} \cdot a_{kk} ~\forall j\neq k$$
Suppose now that $$|a_{jk}| = \max_{x,y \in \{1, \ldots, N\}^2} |a_{xy}|$$
with $j \neq k$.
Since $|a_{jk}|^2 <  a_{jj} \cdot a_{kk} \forall j \neq k$ then one of the following is true:


*

*$a_{jj} < |a_{jk}| < a_{kk}$

*$a_{kk} < |a_{jk}| < a_{jj}$


In the first case, we get a contradiction since $|a_{jk}| < a_{kk}$. Even the second case yields to a similar contradiction since $|a_{jk}| < a_{jj}$. 
This means that we can't have a maximum with $j \neq k$ and hence 
$$|a_{kk}| = \max_{x,y \in \{1, \ldots, N\}^2} |a_{xy}|$$
for some $k$.
