Hessian Matrix
A square matrix of second-order partial derivatives of a scalar-valued function, describing local surface curvature and local extrema stability.
Governing formula
H_ij = ∂²f / (∂x_i ∂x_j)
SI unit
Second Derivative Matrix
In depth
Used in multivariable optimization and stability analysis. Positive definite Hessian matrices correspond to local energy minima (stable equilibrium), while mixed signs indicate saddle points.
Examples in the real world
Evaluating potential energy surface curvature around equilibrium positions to determine normal mode vibration frequencies.