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In mathematics, the Hessian matrix, Hessian or (less commonly) Hesse matrix is a square matrix of secondorder partial derivatives of a scalarvalued 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".
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Transcription
Definitions and properties
Suppose is a function taking as input a vector and outputting a scalar If all secondorder partial derivatives of exist, then the Hessian matrix of is a square matrix, usually defined and arranged as
If furthermore the second partial derivatives are all continuous, the Hessian matrix is a symmetric matrix by the symmetry of second derivatives.
The determinant of the Hessian matrix is called the Hessian determinant.^{[1]}
The Hessian matrix of a function is the transpose of the Jacobian matrix of the gradient of the function ; that is:
Applications
Inflection points
If is a homogeneous polynomial in three variables, the equation is the implicit equation of a plane projective curve. The inflection points of the curve are exactly the nonsingular points where the Hessian determinant is zero. It follows by Bézout's theorem that a cubic plane curve has at most inflection points, since the Hessian determinant is a polynomial of degree
Secondderivative test
The Hessian matrix of a convex function is positive semidefinite. Refining this property allows us to test whether a critical point is a local maximum, local minimum, or a saddle point, as follows:
If the Hessian is positivedefinite at then attains an isolated local minimum at If the Hessian is negativedefinite at then attains an isolated local maximum at If the Hessian has both positive and negative eigenvalues, then is a saddle point for Otherwise the test is inconclusive. This implies that at a local minimum the Hessian is positivesemidefinite, and at a local maximum the Hessian is negativesemidefinite.
For positivesemidefinite and negativesemidefinite 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 secondderivative 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 is a local minimum, and if it is negative, then is a local maximum; if it is zero, then the test is inconclusive. In two variables, the determinant 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 secondderivative test is inconclusive.
Equivalently, the secondorder conditions that are sufficient for a local minimum or maximum can be expressed in terms of the sequence of principal (upperleftmost) minors (determinants of submatrices) 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 minor being negative.
Critical points
If the gradient (the vector of the partial derivatives) of a function is zero at some point then has a critical point (or stationary point) at The determinant of the Hessian at is called, in some contexts, a discriminant. If this determinant is zero then is called a degenerate critical point of or a nonMorse critical point of Otherwise it is nondegenerate, and called a Morse critical point of
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 principal curvatures of the function, and the eigenvectors are the principal directions of curvature. (See Gaussian curvature § Relation to principal curvatures.)
Use in optimization
Hessian matrices are used in largescale optimization problems within Newtontype methods because they are the coefficient of the quadratic term of a local Taylor expansion of a function. That is,
Such approximations may use the fact that an optimization algorithm uses the Hessian only as a linear operator and proceed by first noticing that the Hessian also appears in the local expansion of the gradient:
Letting for some scalar this gives
Notably regarding Randomized Search Heuristics, the evolution strategy's covariance matrix adapts to the inverse of the Hessian matrix, up to a scalar factor and small random fluctuations. This result has been formally proven for a singleparent strategy and a static model, as the population size increases, relying on the quadratic approximation.^{[7]}
Other applications
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). It can be used in normal mode analysis to calculate the different molecular frequencies in infrared spectroscopy.^{[8]} It can also be used in local sensitivity and statistical diagnostics.^{[9]}
Generalizations
Bordered Hessian
A bordered Hessian is used for the secondderivative test in certain constrained optimization problems. Given the function considered previously, but adding a constraint function such that the bordered Hessian is the Hessian of the Lagrange function ^{[10]}
If there are, say, constraints then the zero in the upperleft corner is an block of zeros, and there are border rows at the top and border columns at the left.
The above rules stating that extrema are characterized (among critical points with a nonsingular Hessian) by a positivedefinite or negativedefinite Hessian cannot apply here since a bordered Hessian can neither be negativedefinite nor positivedefinite, as if is any vector whose sole nonzero entry is its first.
The second derivative test consists here of sign restrictions of the determinants of a certain set of submatrices of the bordered Hessian.^{[11]} Intuitively, the constraints can be thought of as reducing the problem to one with free variables. (For example, the maximization of subject to the constraint can be reduced to the maximization of without constraint.)
Specifically, sign conditions are imposed on the sequence of leading principal minors (determinants of upperleftjustified submatrices) of the bordered Hessian, for which the first leading principal minors are neglected, the smallest minor consisting of the truncated first rows and columns, the next consisting of the truncated first rows and columns, and so on, with the last being the entire bordered Hessian; if is larger than then the smallest leading principal minor is the Hessian itself.^{[12]} There are thus minors to consider, each evaluated at the specific point being considered as a candidate maximum or minimum. A sufficient condition for a local maximum is that these minors alternate in sign with the smallest one having the sign of A sufficient condition for a local minimum is that all of these minors have the sign of (In the unconstrained case of these conditions coincide with the conditions for the unbordered Hessian to be negative definite or positive definite respectively).
Vectorvalued functions
If is instead a vector field that is,
Generalization to the complex case
In the context of several complex variables, the Hessian may be generalized. Suppose and write Then the generalized Hessian is If satisfies the ndimensional Cauchy–Riemann conditions, then the complex Hessian matrix is identically zero.
Generalizations to Riemannian manifolds
Let be a Riemannian manifold and its LeviCivita connection. Let be a smooth function. Define the Hessian tensor by
See also
 The determinant of the Hessian matrix is a covariant; see Invariant of a binary form
 Polarization identity, useful for rapid calculations involving Hessians.
 Jacobian matrix – Matrix of all firstorder partial derivatives of a vectorvalued function
 Hessian equation
Notes
 ^ Binmore, Ken; Davies, Joan (2007). Calculus Concepts and Methods. Cambridge University Press. p. 190. ISBN 9780521775410. OCLC 717598615.
 ^ Callahan, James J. (2010). Advanced Calculus: A Geometric View. Springer Science & Business Media. p. 248. ISBN 9781441973320.
 ^ Casciaro, B.; Fortunato, D.; Francaviglia, M.; Masiello, A., eds. (2011). Recent Developments in General Relativity. Springer Science & Business Media. p. 178. ISBN 9788847021136.
 ^ Domenico P. L. Castrigiano; Sandra A. Hayes (2004). Catastrophe theory. Westview Press. p. 18. ISBN 9780813341262.
 ^ Nocedal, Jorge; Wright, Stephen (2000). Numerical Optimization. Springer Verlag. ISBN 9780387987934.
 ^ Pearlmutter, Barak A. (1994). "Fast exact multiplication by the Hessian" (PDF). Neural Computation. 6 (1): 147–160. doi:10.1162/neco.1994.6.1.147. S2CID 1251969.
 ^ Shir, O.M.; A. Yehudayoff (2020). "On the covarianceHessian relation in evolution strategies". Theoretical Computer Science. Elsevier. 801: 157–174. doi:10.1016/j.tcs.2019.09.002.
 ^ Mott, Adam J.; Rez, Peter (December 24, 2014). "Calculation of the infrared spectra of proteins". European Biophysics Journal. 44 (3): 103–112. doi:10.1007/s0024901410056. ISSN 01757571. PMID 25538002. S2CID 2945423.
 ^ Liu, Shuangzhe; Leiva, Victor; Zhuang, Dan; Ma, Tiefeng; FigueroaZúñiga, Jorge I. (March 2022). "Matrix differential calculus with applications in the multivariate linear model and its diagnostics". Journal of Multivariate Analysis. 188: 104849. doi:10.1016/j.jmva.2021.104849.
 ^ Hallam, Arne (October 7, 2004). "Econ 500: Quantitative Methods in Economic Analysis I" (PDF). Iowa State.
 ^ Neudecker, Heinz; Magnus, Jan R. (1988). Matrix Differential Calculus with Applications in Statistics and Econometrics. New York: John Wiley & Sons. p. 136. ISBN 9780471915164.
 ^ Chiang, Alpha C. (1984). Fundamental Methods of Mathematical Economics (Third ed.). McGrawHill. p. 386. ISBN 9780070108134.
Further reading
 Lewis, David W. (1991). Matrix Theory. Singapore: World Scientific. ISBN 9789810206895.
 Magnus, Jan R.; Neudecker, Heinz (1999). "The Second Differential". Matrix Differential Calculus : With Applications in Statistics and Econometrics (Revised ed.). New York: Wiley. pp. 99–115. ISBN 047198633X.
External links
 "Hessian of a function", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
 Weisstein, Eric W. "Hessian". MathWorld.