ヘッセ行列
H ( f ) = ∇ ⊗ ∇ f = [ ∂ 2 f ∂ x 1 2 ∂ 2 f ∂ x 1 ∂ x 2 ⋯ ∂ 2 f ∂ x 1 ∂ x n ∂ 2 f ∂ x 2 ∂ x 1 ∂ 2 f ∂ x 2 2 ⋯ ∂ 2 f ∂ x 2 ∂ x n ⋮ ⋮ ⋱ ⋮ ∂ 2 f ∂ x n ∂ x 1 ∂ 2 f ∂ x n ∂ x 2 ⋯ ∂ 2 f ∂ x n 2 ] {\displaystyle H(f)=\nabla \otimes \nabla f={\begin{bmatrix}{\cfrac {\partial ^{2}f}{\partial x_{1}^{2}}}&{\cfrac {\partial ^{2}f}{\partial x_{1}\partial x_{2}}}&\cdots &{\cfrac {\partial ^{2}f}{\partial x_{1}\partial x_{n}}}\\[16pt]{\cfrac {\partial ^{2}f}{\partial x_{2}\partial x_{1}}}&{\cfrac {\partial ^{2}f}{\partial x_{2}^{2}}}&\cdots &{\cfrac {\partial ^{2}f}{\partial x_{2}\partial x_{n}}}\\[16pt]\vdots &\vdots &\ddots &\vdots \\[16pt]{\cfrac {\partial ^{2}f}{\partial x_{n}\partial x_{1}}}&{\cfrac {\partial ^{2}f}{\partial x_{n}\partial x_{2}}}&\cdots &{\cfrac {\partial ^{2}f}{\partial x_{n}^{2}}}\end{bmatrix}}}
である。上記の行列の行列式はヘッシアン (Hessian) と呼ばれる。
ある種の制限つき最適化問題の判定に境界つきヘッセ行列 (bordered Hessian) が利用される。与えられた関数 f(x1, x2, ..., xn) に
g ( x 1 , x 2 , ... , x n ) = c {\displaystyle g(x_{1},x_{2},\dots ,x_{n})=c}
のような制約関数を付け加えて得られる境界つきヘッセ行列とは