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Hessian


The Jacobian of the derivatives partialf/partialx_1, partialf/partialx_2, ..., partialf/partialx_n of a function f(x_1,x_2,...,x_n) with respect to x_1, x_2, ..., x_n is called the Hessian (or Hessian matrix) H of f, i.e.,

 Hf(x_1,x_2,...,x_n)=[(partial^2f)/(partialx_1^2) (partial^2f)/(partialx_1partialx_2) (partial^2f)/(partialx_1partialx_3) ... (partial^2f)/(partialx_1partialx_n); (partial^2f)/(partialx_2partialx_1) (partial^2f)/(partialx_2^2) (partial^2f)/(partialx_2partialx_3) ... (partial^2f)/(partialx_2partialx_n); | | | ... |; (partial^2f)/(partialx_npartialx_1) (partial^2f)/(partialx_npartialx_2) (partial^2f)/(partialx_npartialx_3) ... (partial^2f)/(partialx_n^2).]

As in the case of the Jacobian, the term "Hessian" unfortunately appears to be used both to refer to this matrix and to the determinant of this matrix (Gradshteyn and Ryzhik 2000, p. 1069).

In the second derivative test for determining extrema of a function f(x,y), the discriminant D is given by

 Hf(x,y)=|(partial^2f)/(partialx^2) (partial^2f)/(partialxpartialy); (partial^2f)/(partialypartialx) (partial^2f)/(partialy^2)|.

The Hessian can be implemented in the Wolfram Language as

  HessianH[f_, x_List?VectorQ] := D[f, {x, 2}]

See also

Jacobian, Second Derivative Test

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References

Gradshteyn, I. S. and Ryzhik, I. M. "Hessian Determinants." §14.314 in Tables of Integrals, Series, and Products, 6th ed. San Diego, CA: Academic Press, p. 1069, 2000.

Referenced on Wolfram|Alpha

Hessian

Cite this as:

Weisstein, Eric W. "Hessian." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Hessian.html

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