Hessian Matrix Calculator
Compute the 2×2 Hessian matrix of second partial derivatives for f(x,y) polynomials.
About Hessian Matrix Calculator
The Hessian matrix H contains all second partial derivatives of f(x,y). For a 2D function: H = [[∂²f/∂x², ∂²f/∂x∂y], [∂²f/∂y∂x, ∂²f/∂y²]]. At a critical point (where ∇f = 0), det(H) > 0 with fxx > 0 indicates a local minimum, det(H) > 0 with fxx < 0 a local maximum, and det(H) < 0 a saddle point.
Formulas
Hessian matrix
H = [[fxx, fxy], [fxy, fyy]]
- fxx
- ∂²f/∂x²
- fxy
- ∂²f/∂x∂y
- fyy
- ∂²f/∂y²
Determinant test
det(H) = fxx * fyy - fxy²
- det(H)
- Hessian determinant
Worked example
With the default values below, Hessian Matrix Calculator returns ∂²f/∂x² of 6.000005. Change any input above and every figure here is recalculated in your browser.
Inputs used
- f(x, y)
- x^3 + x*y^2 - 2*y^3
- x value
- 1
- y value
- 1
Results
- ∂²f/∂x²
- 6.000005
- ∂²f/∂x∂y
- 2.000000
- ∂²f/∂y²
- -10.000003
- det(H)
- -64.000066
- Critical point type
Step by step
- 1Values used
- 2Hessian matrix
- 3Determinant test
- 4∂²f/∂x²= 6.000005
- 5∂²f/∂x∂y= 2.000000
- 6∂²f/∂y²= -10.000003
- 7det(H)= -64.000066
Assumptions and limits
- The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 6 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
Frequently Asked Questions
When can I classify a critical point?
Only when the gradient is zero at that point. The second derivative test uses the Hessian determinant and fxx to classify minima, maxima, and saddle points.
What if the determinant is zero?
The test is inconclusive — the point may be a degenerate saddle, flat region, or higher-order extremum requiring further analysis.