Hesse Matrix Calculator – Loan calculator

Hessian Matrix Calculator (2×2)

Determine local extrema and saddle points using the Second Derivative Test.

f_xx (Second partial derivative w.r.t x):

f_yy (Second partial derivative w.r.t y):

f_xy (Mixed partial derivative):

Calculation Results

Determinant (D):

Matrix Structure:

            [  ,  ]

            [  ,  ]

Classification:

Understanding the Hessian (Hesse) Matrix

In mathematics, the Hessian matrix (or Hesse matrix) is a square matrix of second-order partial derivatives of a scalar-valued function. It is a fundamental tool used in multi-variable calculus for optimization and identifying the nature of critical points.

The 2×2 Hessian Matrix Formula

For a function $f(x, y)$, the Hessian matrix is defined as:

            H = | fxx  fxy |

                | fyx  fyy |

Note: According to Clairaut's Theorem, if the second derivatives are continuous, then $f_{xy} = f_{yx}$.

The Second Derivative Test

To classify a critical point $(a, b)$, we calculate the determinant D of the Hessian matrix:

D = (f_xx * f_yy) – (f_xy)²

If D > 0 and f_xx > 0: The point is a Local Minimum.
If D > 0 and f_xx < 0: The point is a Local Maximum.
If D < 0: The point is a Saddle Point.
If D = 0: The test is Inconclusive.

Practical Example

Consider the function f(x, y) = x² + y². Let's analyze it at the point (0, 0):

Find first derivatives: f_x = 2x, f_y = 2y.
Find second derivatives: f_xx = 2, f_yy = 2, f_xy = 0.
Calculate Determinant: D = (2 * 2) – (0)² = 4.
Since D > 0 and f_xx = 2 (which is > 0), the point (0,0) is a Local Minimum.