Hessian/Hesse Matrix - 𝐻

• is a square and symmetric matrix of second-order partial derivatives of a scalar-valued function / 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, ..., 𝑥_𝑘) is defined as:

• 𝐻(𝑓)(𝑥₁, ..., 𝑥_𝑘)[𝑖,𝑗] = (𝛿/𝛿𝑥_𝑖𝛿𝑥_𝑗) 𝑓(𝑥₁, ..., 𝑥_𝑘) # for 𝑖,𝑗 = 1 to 𝑘

Anywhere that the second partial derivatives are continuous, the differential operators are commutative:

• (𝛿/𝛿𝑥_𝑖𝛿𝑥_𝑗) 𝑓(𝑥₁, ..., 𝑥_𝑘) = (𝛿/𝛿𝑥_𝑗𝛿𝑥_𝑖) 𝑓(𝑥₁, ..., 𝑥_𝑘)
  

This implies that 𝐻[𝑖,𝑗] = 𝐻[𝑗,𝑖] so the Hessian matrix is a symmetric matrix