Hessian Matrix
The Hessian matrix of is , a symmetric matrix since ; it supplies the quadratic term of the multivariate Taylor expansion.
Definition 9.3 (Hessian Matrix)
Role in the multivariate Taylor expansion
Writing , the second-order term of the multivariate Taylor expansion of about is a quadratic form in the Hessian matrix:
In coordinate-independent form,
so near a stationary point , where , the shape of is governed by the Hessian alone.
Degenerate stationary point
Consider , which has a (global) minimum at since for all . Here
and at the stationary point the Hessian matrix is
Its eigenvalues are and , so the Hessian is positive semi-definite. The quadratic terms do not classify the stationary point, and higher order terms of the Taylor series are needed.
Related
Stated in
- Definition 9.3 (Hessian Matrix)§9.3.1 Taylor Series for Multivariate Functions