Is a square zero matrix positive semidefinite?

Question

Does the fact that a square zero matrix contains non-negative eigenvalues (zeros) make it proper to say it is positive semidefinite?

Answer 1

The $n \times n$ zero matrix is positive semidefinite and negative semidefinite.

Answer 2

"When in doubt, go back to the basic definitions"! The definition of "positive semi-definite" is "all eigen-values are non-negative". The eigenvalues or the zero matrix are all 0 so, yes, the zero matrix is positive semi-definite. And, as Gary Moon said, it is also negative semi-definite.