I'm reading the following example on Heath's Scientific Computing (page 266, second edition if anyone has it).
"Minimize $f(x_1,x_2)=2\pi x_1(x_1+x_2)$ subject to $g(x_1,x_2)=\pi x^2_1x_2-V$" (minimize the surface area of a cylinder subject to a constant of its volume).
Using Lagrange Multipliers and assuming $V=1000cm^3$, the approximate solutions are $x_1=5.4$, $x_2=10.8$ and $\lambda = -0.37$.
Then $H_f(x)=2\pi \begin{bmatrix} 2 & 1 \\ 1 & 0 \\\end{bmatrix}$ and $H_g(x)=2\pi \begin{bmatrix} x_2 & x_1 \\ x_1 & 0 \\\end{bmatrix}$, so that $B(x^*, \lambda ^*)=2\pi \begin{bmatrix} 2 +x_2\lambda & 1+x_1\lambda \\ 1x_1\lambda & 0 \\\end{bmatrix}$.
Computing we get $B(x^*, \lambda ^*)=2\pi \begin{bmatrix} -12.6 & -6.3 \\ -6.3 & 0 \\\end{bmatrix}$.
Now here comes the part that I don't get
"The matrix is not positive definite, but that is to be expected. To consider the appropiate subspace we compute a null vector $z$ (i.e., a basis for the one dimensional null space) for $J_g(x^*)= \begin{bmatrix} 369 & 92.3 \end{bmatrix}$ and obtain $z=\begin{bmatrix} -0.243 & 0.970 \end{bmatrix}^T$, so that $z^TBz=2.23$, which is positive confirming that the point found is indeed a constrained minimum. The optimal value is $f(x^*)=554cm^2$ of surfaced area".
How can I compute $z$?. What is it?. Sadly, I think I haven't got that part of the theory of constrained optimization.