= Reduced Hessian
{title2=$Z^THZ,\quad\operatorname{range}Z=\ker A$}
For equality-constrained optimization, restrict the <Hessian matrix> $H$ of the <optimization Lagrangian> to feasible tangent directions. If the columns of $Z$ form a <basis> of the nullspace of the constraint <Jacobian matrix> $A$, the resulting quadratic form has matrix $Z^THZ$. Positive definiteness gives positive quadratic curvature in every nonzero tangent direction. Nonsingularity of this reduced matrix, together with full row rank of $A$, characterizes nonsingularity of the corresponding equality-constrained <saddle-point matrix>.
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