Saddle-point matrix
= Saddle-point matrix
{title2=$\begin{pmatrix}H&A^T\\A&-C\end{pmatrix}$}
A common symmetric saddle-point system couples primal unknowns with constraint multipliers through the displayed <block matrix>, with symmetric $H,C$. At $C=0$, full row rank of $A$ and a nonsingular <reduced Hessian> are exactly the nonsingularity conditions. Such matrices arise from equality-constrained quadratic models and mixed <finite element methods>. A regularized block $C=\mu I$ also arises when an augmented linear system replaces a badly scaled quadratic-penalty <Newton system>.