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- Hessian matrix

In mathematics, the **Hessian matrix** or **Hessian** is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematician Ludwig Otto Hesse and later named after him. Hesse originally used the term "functional determinants".

Suppose

*f**:**\R*^{n}*\to**\R*

x*\in**\R*^{n}

*f(x)**\in**\R.*

*f*

H

*f*

*n* x *n*

The Hessian matrix is a symmetric matrix, since the hypothesis of continuity of the second derivatives implies that the order of differentiation does not matter (Schwarz's theorem).

The determinant of the Hessian matrix is called the .^{[1]}

The Hessian matrix of a function

*f*

*f*

H*(f(x))*=J*(\nabla**f(x)).*

If

*f*

*f*=0

9

3*.*

See main article: Second partial derivative test.

*x*

If the Hessian is positive-definite at

*x,*

*f*

*x.*

*x,*

*f*

*x.*

*x*

*f.*

For positive-semidefinite and negative-semidefinite Hessians the test is inconclusive (a critical point where the Hessian is semidefinite but not definite may be a local extremum or a saddle point). However, more can be said from the point of view of Morse theory.

The second-derivative test for functions of one and two variables is simpler than the general case. In one variable, the Hessian contains exactly one second derivative; if it is positive, then

*x*

*x*

Equivalently, the second-order conditions that are sufficient for a local minimum or maximum can be expressed in terms of the sequence of principal (upper-leftmost) minors (determinants of sub-matrices) of the Hessian; these conditions are a special case of those given in the next section for bordered Hessians for constrained optimization—the case in which the number of constraints is zero. Specifically, the sufficient condition for a minimum is that all of these principal minors be positive, while the sufficient condition for a maximum is that the minors alternate in sign, with the

1 x 1

If the gradient (the vector of the partial derivatives) of a function

*f*

x*,*

*f*

x*.*

x

x

*f,*

*f.*

*f.*

The Hessian matrix plays an important role in Morse theory and catastrophe theory, because its kernel and eigenvalues allow classification of the critical points.^{[2]} ^{[3]} ^{[4]}

The determinant of the Hessian matrix, when evaluated at a critical point of a function, is equal to the Gaussian curvature of the function considered as a manifold. The eigenvalues of the Hessian at that point are the principle curvatures of the function, and the eigenvectors are the principle directions of curvature. (See .)

Hessian matrices are used in large-scale optimization problems within Newton-type methods because they are the coefficient of the quadratic term of a local Taylor expansion of a function. That is,$$y\; =\; f(\backslash mathbf\; +\; \backslash Delta\backslash mathbf)\backslash approx\; f(\backslash mathbf)\; +\; \backslash nabla\; f(\backslash mathbf)\; \backslash Delta\backslash mathbf\; +\; \backslash frac\; \backslash ,\; \backslash Delta\backslash mathbf^\backslash mathrm\; \backslash mathbf(\backslash mathbf)\; \backslash ,\; \backslash Delta\backslash mathbf$$where

*\nabla**f*

\left( | \partialf |

\partialx_{1} |

*,**\ldots,*

\partialf | |

\partialx_{n} |

*\right).*

*\Theta\left(n*^{2\right)}

H*(v),*

Letting

*\Delta*x=*r*v

*r,*

*r*

l{O}(r)

The Hessian matrix is commonly used for expressing image processing operators in image processing and computer vision (see the Laplacian of Gaussian (LoG) blob detector, the determinant of Hessian (DoH) blob detector and scale space). The Hessian matrix can also be used in normal mode analysis to calculate the different molecular frequencies in infrared spectroscopy.^{[7]}

A is used for the second-derivative test in certain constrained optimization problems. Given the function

*f*

*g*

*g(x)*=*c,*

Λ*(x,*λ*)*=*f(x)*+λ*[g(x)*-*c]:*

If there are, say,

*m*

*m* x *m*

*m*

*m*

The above rules stating that extrema are characterized (among critical points with a non-singular Hessian) by a positive-definite or negative-definite Hessian cannot apply here since a bordered Hessian can neither be negative-definite nor positive-definite, as

z^{T}Hz=0

z

The second derivative test consists here of sign restrictions of the determinants of a certain set of

*n*-*m*

*m*

*n*-*m*

*f\left(x*_{1,}*x*_{2,}*x*_{3\right)}

*x*_{1}+*x*_{2}+*x*_{3}=1

*f\left(x*_{1,}*x*_{2,}1-*x*_{1}-*x*_{2\right)}

Specifically, sign conditions are imposed on the sequence of leading principal minors (determinants of upper-left-justified sub-matrices) of the bordered Hessian, for which the first

2*m*

2*m*+1

2*m*+2

2*m*+1

*n*+*m,*

*n*-*m*

*(*-1*)*^{m+1}*.*

*(*-1*)*^{m.}

*m*=0

If

*f*

f*:**\R*^{n}*\to**\R*^{m,}

*n* x *n*

*m*

f

*m*=1*.*

In the context of several complex variables, the Hessian may be generalized. Suppose

*f**:**\Complex*^{n}*\to**\Complex,*

*f\left(z*_{1,}*\ldots,**z*_{n\right).}

\partial^{2f} | |

\partialz_{i}\partial\overline{z_{j} |

*f*

Let

*(M,g)*

*\nabla*

*f**:**M**\to**\R*

*\left\{x*^{i\right\}}

k | |

\Gamma | |

ij |

- The determinant of the Hessian matrix is a covariant; see Invariant of a binary form
- Polarization identity, useful for rapid calculations involving Hessians.

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