Hessian Matrix

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Hessian Matrix

f

if all partial second derivatives of ''f'' exist, then the Hessian matrix of f is the matrix

:H(''f'')_''ij''(''x'') = ''D''_''i'' ''D''_''j'' ''f''(''x'')

where ''x'' = (''x''₁, ''x''₂, ..., ''x''_''n''). That is,

:

H(f) = \begin{bmatrix}
rac{\partial^2 f}{\partial x_1^2} & rac{\partial^2 f}{\partial x_1\partial x_2} & \cdots & rac{\partial^2 f}{\partial x_1\partial x_n} \
rac{\partial^2 f}{\partial x_2\partial x_1} & rac{\partial^2 f}{\partial x_2^2} & \cdots & rac{\partial^2 f}{\partial x_2\partial x_n} \
dots & dots & \ddots & dots \
rac{\partial^2 f}{\partial x_n\partial x_1} & rac{\partial^2 f}{\partial x_n\partial x_2} & \cdots & rac{\partial^2 f}{\partial x_n^2}
\end{bmatrix}

The term Hessian was coined by James Joseph Sylvester , named for German mathematician Ludwig Otto Hesse , who had used the term functional determinants.

MIXED DERIVATIVES AND SYMMETRY OF THE HESSIAN

The mixed derivatives of ''f'' are the entries ''off'' the Main Diagonal in the Hessian. Here, the order of differentiation does not matter. For example,

:

rac {\partial}{\partial x} \left( rac { \partial f }{ \partial y} 
ight) =
       rac {\partial}{\partial y} \left( rac { \partial f }{ \partial x} 
ight)

This can also be written as:

:

f_{xy} = f_{yx}

In a formal statement: if the second derivatives of ''f'' are all Continuous in a region ''D'', then the Hessian of ''f'' is a Symmetric Matrix throughout ''D''; see Symmetry Of Second Derivatives .

CRITICAL POINTS AND DISCRIMINANT

If the Gradient of ''f'' (i.e. its derivative in the vector sense) is zero at some point ''x'', then ''f'' has a '' Critical Point '' at ''x''. The Determinant of the Hessian at ''x'' is then called the Discriminant . If this determinant is zero then ''x'' is called a ''degenerate critical point'' of ''f''. Otherwise it is not degenerate.

SECOND DERIVATIVE TEST

The following test can be applied at a non-degenerate critical point ''x''. If the Hessian is Positive Definite at x, then ''f'' attains a local minimum at ''x''. If the Hessian is negative definite at x, then ''f'' attains a local maximum at ''x''. If the Hessian has both positive and negative Eigenvalue s then ''x'' is a Saddle Point for ''f'' (this is true even if ''x'' is degenerate). Otherwise the test is inconclusive.

Note that for positive semidefinite and negative semidefinite Hessians the test is inconclusive. However, more can be said from the point of view of Morse Theory .

In view of what has just been said, the Second Derivative Test for functions of one and two variables are simple. In one variable, the Hessian contains just one second derivative; if it's positive then ''x'' is a local minimum, if it's negative then ''x'' is a local maximum; if it's zero then the test is inconclusive. In two variables, the discriminant can be used, because the determinant is the product of the eigenvalues. If it is positive then the eigenvalues are both positive, or both negative. If it is negative then the two eigenvalues have different signs. If it is zero, then the second derivative test is inconclusive.

VECTOR-VALUED FUNCTIONS

If ''f'' is instead vector-valued, i.e.

f

then the array of second partial derivatives is not a matrix, but a Tensor of rank 3.