Editing Hessian matrix (section)

=== Critical points ===

If the [[gradient]] (the vector of the partial derivatives) of a function <math>f</math> is zero at some point <math>\mathbf{x},</math> then <math>f</math> has a {{em|[[Critical point (mathematics)|critical point]]}} (or {{em|[[stationary point]]}}) at <math>\mathbf{x}.</math> The [[determinant]] of the Hessian at <math>\mathbf{x}</math> is called, in some contexts, a [[discriminant]]. If this determinant is zero then <math>\mathbf{x}</math> is called a {{em|degenerate critical point}} of <math>f,</math> or a {{em|non-Morse critical point}} of <math>f.</math> Otherwise it is non-degenerate, and called a {{em|Morse critical point}} of <math>f.</math>

The Hessian matrix plays an important role in [[Morse theory]] and [[catastrophe theory]], because its [[Kernel of a matrix|kernel]] and [[eigenvalue]]s allow classification of the critical points.<ref>{{Cite book|url=https://books.google.com/books?id=geruGMKT9_UC&pg=PA248|title=Advanced Calculus: A Geometric View|last=Callahan|first=James J.|date=2010|publisher=Springer Science & Business Media|isbn=978-1-4419-7332-0|page=248|language=en}}</ref><ref>{{Cite book|url=https://books.google.com/books?id=Tcn3CAAAQBAJ&pg=PA178|title=Recent Developments in General Relativity|editor-last=Casciaro|editor-first=B.|editor-last2=Fortunato|editor-first2=D.|editor-last3=Francaviglia|editor-first3=M.|editor-last4=Masiello|editor-first4=A.|date=2011|publisher=Springer Science & Business Media|isbn=9788847021136|page=178|language=en}}</ref><ref>{{cite book|author1=Domenico P. L. Castrigiano|author2=Sandra A. Hayes|title=Catastrophe theory|year=2004|publisher=Westview Press|isbn=978-0-8133-4126-2|page=18}}</ref>

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 {{section link|Gaussian curvature|Relation to principal curvatures}}.)