 Bipartite block-constant positive semidefinite matrix

For positive block sizes, the real matrix $\left(\begin{array}{cc}aJ& bJ\\ bJ& aJ\end{array}\right)$ is a positive semidefinite matrix exactly when $a\ge |b|$. For test-vector block sums $s,t$, its quadratic form is $(a+b)(s+t{)}^2/2+(a-b)(s-t{)}^2/2$. This proves sufficiency; testing $s=\pm t$ proves necessity. Zero sums in both blocks describe part of its kernel.