A is a matrix of positive defined quadratic form. How can I show, $A^{-1}$ is the same? Let a square matrix A is a matrix of positive defined quadratic form. How can I show, that $A^{-1}$ also a matrix of a positive defined quadratic form?
Positive defined quadratic form is
A(x,y), that all it's corner minors are positive.
Inverse matrix is a matrix, contains all minors of the source matrix.
It must be linked somehow...

Let $z=Ax$. $A$ is positive definite and hence also invertible. Thus, it follows that $
x=A^{-1}z.
$
Premultiply by $z'$ to get $z'x=z'A^{-1}z$. But, $z'=x'A'$ so what you have is actually
$$
x'A'x=z'A^{-1}z>0
$$
since $A=A'$ and $x'A'x>0$ by the nature of semi definite matrices.
