The Hessian matrix is the matrix of all combinations of second-order derivatives, for example:
$$
H=
\left [
\begin{array}{rr}
\frac{\partial^2 f(x,y)}{\partial x^2}& \frac{\partial^2 f(x,y)}{\partial y\partial x} \\
\frac{\partial^2 f(x,y)}{\partial x\partial y}& \frac{\partial^2 f(x,y)}{\partial y^2}
\end{array}
\right ]
$$

% this is actually (y-x^2)^2+(x-2)^2
\textbf{Example:} Consider the function $f(x,y)=x^4+x^2(1-2y) + y^2 -4x +4$. The gradient of this function at a general point $(x,y)$ is

$$
\nabla f (\mathbf{x} )=\left [ \begin{array}{r}

\frac{\partial f(x,y)}{\partial x_1}\\
\frac{\partial f(x,y)}{\partial x_2}
\end{array}
\right ]=\left [ \begin{array}{r}
4x^3+2x(1-2y)-4\\
2y-2x^2
\end{array}
\right ]
$$

Hence e.g. at $(x,y)=(0,1)$ we can calculate the gradient at this particular point as
$$
\nabla f (\mathbf{x} )=\left [ \begin{array}{r}
-4\\
2
\end{array}
\right ]
$$
and the Hessian is
$$
H=
\left [
\begin{array}{rr}
\frac{\partial^2 f(x,y)}{\partial x^2}& \frac{\partial^2 f(x,y)}{\partial y\partial x} \\
\frac{\partial^2 f(x,y)}{\partial x\partial y}& \frac{\partial^2 f(x,y)}{\partial y^2}
\end{array}
\right ]
=
\left [
\begin{array}{rr}
12x^2+2(1-2y)&-4x \\
-4x&2
\end{array}
\right ]
$$
so e.g. at the point $(x,y)=(0,1)$ the value of the Hessian is ...

The Hessian matrix is the matrix of all combinations of second-order derivatives, for example:
$$
H=
\left [
\begin{array}{rr}
\frac{\partial^2 f(x,y)}{\partial x^2}& \frac{\partial^2 f(x,y)}{\partial y\partial x} \\
\frac{\partial^2 f(x,y)}{\partial x\partial y}& \frac{\partial^2 f(x,y)}{\partial y^2}
\end{array}
\right ]
$$

The Hessian matrix is the matrix of all combinations of second-order derivatives, for example:
$$
H=
\left [
\begin{array}{rr}
\frac{\partial^2 f(x,y)}{\partial x^2}& \frac{\partial^2 f(x,y)}{\partial y\partial x} \\
\frac{\partial^2 f(x,y)}{\partial x\partial y}& \frac{\partial^2 f(x,y)}{\partial y^2}
\end{array}
\right ]
$$

% this is actually (y-x^2)^2+(x-2)^2
\textbf{Example:} Consider the function $f(x,y)=x^4+x^2(1-2y) + y^2 -4x +4$. The gradient of this function at a general point $(x,y)$ is

$$
\nabla f (\mathbf{x} )=\left [ \begin{array}{r}

\frac{\partial f(x,y)}{\partial x_1}\\
\frac{\partial f(x,y)}{\partial x_2}
\end{array}
\right ]=\left [ \begin{array}{r}
4x^3+2x(1-2y)-4\\
2y-2x^2
\end{array}
\right ]
$$

Hence e.g. at $(x,y)=(0,1)$ we can calculate the gradient at this particular point as
$$
\nabla f (\mathbf{x} )=\left [ \begin{array}{r}
-4\\
2
\end{array}
\right ]
$$
and the Hessian is
$$
H=
\left [
\begin{array}{rr}
\frac{\partial^2 f(x,y)}{\partial x^2}& \frac{\partial^2 f(x,y)}{\partial y\partial x} \\
\frac{\partial^2 f(x,y)}{\partial x\partial y}& \frac{\partial^2 f(x,y)}{\partial y^2}
\end{array}
\right ]
=
\left [
\begin{array}{rr}
12x^2+2(1-2y)&-4x \\
-4x&2
\end{array}
\right ]
$$
so e.g. at the point $(x,y)=(0,1)$ the value of the Hessian is ...