Show that all 2x2 sub matrices of a positive definite symmetric matrix are positive semidefinite. I have to show that all $2\times2$ sub matrices
$\left[ \begin{array}{rr}
X_{ii} & X_{ij} \\ 
X_{ij} & X_{jj}  \\
\end{array}\right]$
of a real, positive definite symmetric matrix $X$ are positive semi definite. After that I would like to proof that for this $2\times 2$ sub matrix the inequality $X_{ii}X_{jj}-X_{ij}^{2}≥0$ holds.
Can anybody help me?
I've tried to do so with the definition of positive and positive semi definite matrices (positive definite iff $x^{T}Ax>0$ $\forall x\neq 0 $), but I am desperate.
Thanks for your help in advance.
 A: Can you clarify what you mean by "sub-matrices" more precisely? Perhaps you mean $2 \times 2$ principal minors (i.e. submatrices where we select the same two rows and columns).
Hint 1. For simplicity, suppose that $A$ is the top left $2 \times 2$ principal minor of $X$: consider $z = (z_1, z_2, 0, \dots, 0)$, and observe that
$$
z^T X z = (z_1, z_2)^T A (z_1, z_2) .
$$
What can you say about this? Generalise this to arbitrary $2 \times 2$ (principal minor) submatrices in the obvious way.
Hint 2. Suppose we know that all $2 \times 2$ (principal minor) submatrices are positive semi-definite. Let $A$ be any such submatrix, say
$$
A =
\begin{pmatrix}
a_{11} & a_{12} \\
a_{21} & a_{22} \\
\end{pmatrix} .
$$
Observe that $\det(A) = a_{11} a_{22} - a_{12} a_{21} = a_{11} a_{22} - a_{12}^2$ if $A$ is symmetric (this is another reason why principal minor makes sense). This looks like the expression you want. What do you know about the relationship between the determinant of a matrix, its eigenvalues, and how does this relate to positive semi-definiteness?
P.S. You may want to look up Sylvester's criterion.
