# Is a square zero matrix positive semidefinite?

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

The $$n \times n$$ zero matrix is positive semidefinite and negative semidefinite.
• Nitpicking, the definition of "positive-semidefinite" is "a symmetric matrix $A$ with $\vec x^{\mathsf T}\!A\vec x \ge 0$ for all $\vec x \in \mathbb R^n$." The eigenvalue test is equivalent to this, but is not the definition. – Misha Lavrov Mar 19 at 3:56