【へっせぎょうれつ (Hessian matrix)】
多変数スカラ値関数 f ( x ) = f ( x 1 , ⋯ , x n ) {\displaystyle f(x)=f(x_{1},\cdots ,x_{n})\,} を各変数に関して2階偏微分した2階共変テンソルのこと. 通常 H ( x ) {\displaystyle H({\boldsymbol {x}})\,} と行列で表記する:
H ( x ) := [ ∂ 2 f ∂ x 1 ∂ x 1 ( x ) ⋯ ∂ 2 f ∂ x 1 ∂ x n ( x ) ⋮ ⋮ ∂ 2 f ∂ x n ∂ x 1 ( x ) ⋯ ∂ 2 f ∂ x n ∂ x n ( x ) ] . {\displaystyle H({\boldsymbol {x}}):=\left[{\begin{array}{ccc}{\frac {\partial ^{2}f}{\partial x_{1}\partial x_{1}}}({\boldsymbol {x}})&\cdots &{\frac {\partial ^{2}f}{\partial x_{1}\partial x_{n}}}({\boldsymbol {x}})\\\vdots &&\vdots \\{\frac {\partial ^{2}f}{\partial x_{n}\partial x_{1}}}({\boldsymbol {x}})&\cdots &{\frac {\partial ^{2}f}{\partial x_{n}\partial x_{n}}}({\boldsymbol {x}})\end{array}}\right].}
を各変数に関して2階偏微分した2階共変テンソルのこと. 通常 
と行列で表記する:![{\displaystyle H({\boldsymbol {x}}):=\left[{\begin{array}{ccc}{\frac {\partial ^{2}f}{\partial x_{1}\partial x_{1}}}({\boldsymbol {x}})&\cdots &{\frac {\partial ^{2}f}{\partial x_{1}\partial x_{n}}}({\boldsymbol {x}})\\\vdots &&\vdots \\{\frac {\partial ^{2}f}{\partial x_{n}\partial x_{1}}}({\boldsymbol {x}})&\cdots &{\frac {\partial ^{2}f}{\partial x_{n}\partial x_{n}}}({\boldsymbol {x}})\end{array}}\right].}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/c4e6903bc1102b939150840cfcd90a3227275881)