Hessian Matrix Calculator
Calculate Hessian matrix
Frequently Asked Questions
What is a Hessian matrix?
The Hessian matrix is a square matrix of second-order partial derivatives of a multivariable function. For f(x,y), it is a 2×2 matrix: [[∂²f/∂x², ∂²f/∂x∂y], [∂²f/∂y∂x, ∂²f/∂y²]]. It describes the local curvature of the function.
How is the Hessian used to classify critical points?
At a critical point, compute the Hessian determinant D = f_xx × f_yy - (f_xy)². If D > 0 and f_xx > 0: local minimum. If D > 0 and f_xx < 0: local maximum. If D < 0: saddle point. If D = 0: test is inconclusive.
Where is the Hessian matrix used in practice?
The Hessian is used in optimization algorithms (Newton's method), machine learning (second-order optimization), image processing (blob detection), and economics (analyzing utility and production functions).
What does a positive definite Hessian mean?
A positive definite Hessian (all eigenvalues positive) means the function is locally convex at that point — it curves upward in all directions. This guarantees a local minimum. A negative definite Hessian indicates a local maximum.
How do I compute the Hessian for a 3-variable function?
For f(x,y,z), the Hessian is a 3×3 symmetric matrix with 6 unique entries: f_xx, f_yy, f_zz on the diagonal, and f_xy, f_xz, f_yz off-diagonal. The calculator computes all partial derivatives and assembles the matrix.