Hessian/Hesse Matrix - 𝐻
- is a square and symmetric matrix of second-order partial derivatives of a scalar field
- it describes the local curvature of a function of many variables
- equivalently, the Hessian is the Jacobian of the gradient (However, Jacobian involves a vector-valued function?)
Hessian Matrix - Definition
The Hessian matrix 𝐻(𝑓)(𝑥1, …, 𝑥𝑘) is defined as:
- 𝐻(𝑓)(𝑥1, …, 𝑥𝑘)[𝑖,𝑗] = (𝛿/𝛿𝑥𝑖𝛿𝑥𝑗) 𝑓(𝑥1, …, 𝑥𝑘) # for 𝑖,𝑗 = 1 to 𝑘
Anywhere that the second partial derivatives are continuous, the differential operators are commutative:
- (𝛿/𝛿𝑥𝑖𝛿𝑥𝑗) 𝑓(𝑥1, …, 𝑥𝑘) = (𝛿/𝛿𝑥𝑗𝛿𝑥𝑖) 𝑓(𝑥1, …, 𝑥𝑘)
This implies that 𝐻[𝑖,𝑗] = 𝐻[𝑗,𝑖] so the Hessian matrix is a symmetric matrix