Positive definite matrices and their minors How can I show that a positive definite matrix $A$ has a positive determinant?
More generally, are the minors of a positive definite matrix positive?
 A: Hint: A is positive definite matrix  $\to$ all eigen value of A be  positive 
then we conclude  eigenvalues of a positive definite matrix are real and positive.
and The determinant is the product of the eigenvalues.
 $$
\ detA=\prod_{i=1}^na_i :\text{$a_i$ is eigen value }\to\ detA\gt0
$$ 

!Proof: let A be positive defined matrix and $v$ be eigen vector correspond to $a$ eigen value of $ A$ ($Av=av$) then $\forall 0\neq x :x^tAx\ge0 $
  i know $V\neq0\to v^tAv=v^taV=av^tV=a\sum_{i=1}^nv_i^2a\ge0 \to a\ge0$ 

A: If the matrix $A$ is positive definite then you know that the bilinear form represented by the matrix $A$ is also positive definite.
Now by Sylverster's theorem you know that for each bilinear form there exists a basis where the matrix of the form is diagonal (with only positive, negative or zero entries on the diagonal). For a positive definite form though, there are only positive entries on the diagonal.
So we have two basis for our bilinear form. One where its matrix is $A$ and one where it's matrix is $B$, where $B$ is diagonal and has only positive entries on the diagonal.
So there is an invertible matrix $S$ such that : $A=S^tBS$.
Thus $\det(A)=\det(S^t)\det(B)\det(S)=\det(S)^2\det(B)>0$
