Why Does a 2x2 Matrix [x y; y z] in the PSD Cone Imply x>=0, z>=0, and xz>=y^2?

[peterlam]

Hi!

If we have a 2x2 matrix [x y;y z] belonging to a positive semidefinite cone. Why is it equivalent to say x>=0, z>=0, and xz>=y^2?

Thanks!

 

[ViolentGreen]

A matrix is PSD if and only if it's principal minors are nonnegative (see http://en.wikipedia.org/wiki/Positive-definite_matrix#Characterizations").

A 2x2 matrix has three principal minors - the diagonal elements, and the determinant. So x,z >= 0, and xz - y^2 >=0.

I'm sure there is a way to see this without having to use the principal minors characterization though.