For "outer product" in geometric algebra, see exterior product. In linear algebra, the outer product typically refers to the tensor product of two vectors. The result of applying the outer product to a pair of vectors is a matrix. The name contrasts with the inner product, which takes as input a pair of vectors and produces a scalar. The outer product of vectors can be also regarded as a special case of the Kronecker product of matrices.

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  • For "outer product" in geometric algebra, see exterior product. In linear algebra, the outer product typically refers to the tensor product of two vectors. The result of applying the outer product to a pair of vectors is a matrix. The name contrasts with the inner product, which takes as input a pair of vectors and produces a scalar. The outer product of vectors can be also regarded as a special case of the Kronecker product of matrices. Some authors use the expression "outer product of tensors" as a synonym of "tensor product". The outer product is also a higher-order function in some computer programming languages such as APL and Mathematica.
  • 直積 (ちょくせき、outer product) は、2つのベクトルからテンソルを導き出す演算。スカラー積（内積）・ベクトル積（クロス積）に対しテンソル積とも呼ばれるが、テンソル積はもっと広い意味の用語であり、直積はベクトル同士のテンソル積の1つ（もう1つは内積）である。 直積は、実ベクトル <math> \mathbf a, \mathbf b</math> に対して <math> \mathbf a \circ \mathbf b = \mathbf a \mathbf b^\operatorname{T} = (a_i b_j) = \begin{pmatrix} a_1 b_1 & a_1 b_2 & \cdots & a_1 b_n \\ a_2 b_1 & a_2 b_2 & \cdots & a_2 b_n \\ \vdots & \vdots & \ddots & \vdots \\ a_n b_1 & a_n b_2 & \cdots & a_n b_n \end{pmatrix}</math> と定義される。n は次元数。<math> M ^\operatorname{T} </math> は転置行列で、ベクトルは列行列とみなす。 複素ベクトルに対しては <math> \mathbf a \circ \mathbf b = \mathbf a \mathbf b^* = (a_i \bar{b_j})</math> と定義される。<math> M ^* \,</math> は共役転置行列。 この定義は内積 <math> \mathbf a \cdot \mathbf b = \mathbf a^\operatorname{T} \mathbf b = a_i b_i </math> と対称をなしている。同じ添え字 (i) は縮約記法で、i についての総和を取る。 外積は <math> \mathbf a \wedge \mathbf b = \mathbf a \circ \mathbf b - \mathbf b \circ \mathbf a </math> と表せ、特に3次元でのクロス積は <math> \mathbf a \times \mathbf b = \left(\frac{1}{2} \epsilon_{ijk} _{jk} \right) </math> とも表せる。<math>\epsilon_{ijk}\,</math> はエディントンのイプシロン。
  • 外积典型的称呼张量积或有类似势的运算如楔积。这些运算的势是笛卡尔积的势。 这个名字相对于内积，它是有相反次序的积。
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  • For "outer product" in geometric algebra, see exterior product. In linear algebra, the outer product typically refers to the tensor product of two vectors. The result of applying the outer product to a pair of vectors is a matrix. The name contrasts with the inner product, which takes as input a pair of vectors and produces a scalar. The outer product of vectors can be also regarded as a special case of the Kronecker product of matrices.
  • 外积典型的称呼张量积或有类似势的运算如楔积。这些运算的势是笛卡尔积的势。 这个名字相对于内积，它是有相反次序的积。
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  • Outer product
  • 直積 (ベクトル)
  • 外积
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