[Math] Show that 2×2 matrix (1 1 1 1) is symmetric positive semidefinite but is not symmetric positive definite

Question

Should one calculate the eigenvalues and find that one of them is positive and the other is 0? Theorem says that if all eigenvalues are greater of equal to zero then matrix is spsd. But if all eigenvalues are not strictly greater than 0 then matrix is not spd.

Answer 1

You can simply check the associated quadratic form is $(x+y)^2$. It's clearly positive, not definite.