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A Hessian matrix or simply a Hessian is a matrix of all the second-order partial derivatives of a function . For example, given the function The resulting Hessian is 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.

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  • Hessian matrix
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  • A Hessian matrix or simply a Hessian is a matrix of all the second-order partial derivatives of a function . For example, given the function The resulting Hessian is 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.
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  • A Hessian matrix or simply a Hessian is a matrix of all the second-order partial derivatives of a function . For example, given the function The resulting Hessian is 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 . ∇ is the del operator and T represents the transpose. If there are more constraints, more columns and rows can be added. File:Helicoid.svg This multivariable calculus-related article contains minimal information concerning its topic. You can help the Mathematics Wikia by adding to it.
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