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A Hessian matrix or simply a Hessian is a matrix of all the second-order partial derivatives of a function f:\R^n\to\R . For example, given the function

f(x,y)=4xy+y^2

The resulting Hessian is

\bold{H}(f)=\begin{bmatrix}0&4+2y\\4+2y&2\end{bmatrix}

The Hessian matrix will be symmetric if the partial derivatives of the function are continuous.

The determinant of a Hessian matrix can be used as a generalisation of the second derivative test for single-variable functions. If the determinant of the Hessian positive, it will be an extreme value (minimum if the matrix is positive definite). If it is negative, there will be a saddle point. If it is 0, another test must be used.

The Hessian can be thought of as the second derivative of a multivariable function, with gradient being the first and higher order derivatives being tensors of higher rank.

A bordered Hessian is a similar matrix used to optimize a multivariable function with a constraint g(x_1,\ldots,x_n)=c .


\bold{H}(f,g)=\begin{bmatrix}
0&\nabla g\\\nabla g^{\rm T}&\bold{H}(f)\end{bmatrix}
\begin{bmatrix}
0&\dfrac{\part g}{\part x_1}&\cdots&\dfrac{\part g}{\part x_n}\\[2.2ex]
\dfrac{\part g}{\part x_1}&\dfrac{\part^2f}{\part x_1^2}&\cdots&\dfrac{\part^2f}{\part x_1\part x_n}\\[2.2ex]
\vdots&\vdots&\ddots&\vdots\\[2.2ex]
\dfrac{\part g}{\part x_n}&\dfrac{\part^2f}{\part x_n\part x_1}&\cdots&\dfrac{\part^2f}{\part x_n^2}
\end{bmatrix}

∇ is the del operator and T represents the transpose. If there are more constraints, more columns and rows can be added.

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