This section is divided into a number of subsections, links to which are:

Wedge product

The wedge product of two vectors u and v measures the noncommutativity of their tensor product.

An wedge product (also known as exterior product) is the tensor product of two vectors

${\bf u} = \left[ u_1 , u_2 , \ldots , u_n \right]$ and

${\bf v} = \left[ v_1 , v_2 , \ldots , v_n \right] ,$ denoted u ∧ v, is the square matrix defined by

${\bf v} \wedge {\bf u} = {\bf v} \otimes {\bf u} - {\bf u} \otimes {\bf v} .$ Equivalently,

$\left( {\bf v} \wedge {\bf u} \right)_{ij} = \left( v_i u_j - v_j u_i \right) .$

Example 1: We are going to show that the wedge product u ∧ v has, up to sign, six, not four, distinct entries. Let u = [ 𝑎, b, c, d ] and v = [ α, β, γ, δ ] be any two vectors in ℝ⁴. Their wedge product is the square matrix:

${\bf u} \wedge {\bf v} = \begin{bmatrix} 0 & a\beta - \alpha b & a\gamma - \alpha c & a\delta - \alpha d \\ b \alpha - \beta a & 0 & b \gamma - \beta c & b \delta - \beta d \\ c \alpha - \gamma a & c \beta - \gamma b & 0 & c \delta - \gamma d \\ d \alpha - \delta a & d \beta - \delta b & d \gamma - \delta c & 0 \end{bmatrix} .$

Let us introduce six parameters:

$c_1 = a\beta - \alpha b , \quad c_2 = a\gamma - \alpha c , \quad c_3 = a\delta - \alpha d ,$

and

$c_4 = b \gamma - \beta c , \quad c_5 = b \delta - \beta d , \quad c_6 = c \delta - \gamma d .$

Then the wedge product of these two vectors becomes

${\bf u} \wedge {\bf v} = \begin{bmatrix} 0 & c_1 & c_2 & c_3 \\ -c_1 & 0 & c_4 & c_5 \\ - c_2 & - c_4 & 0 & c_6 \\ - c_3 & - c_5 & - c_6 & 0 \end{bmatrix} .$

End of Example 4

It can be shown that in n dimensions, the antisymmetric matrix u ∧ v has n(n − 1)/2 unique entries. The wedge product is an antisymmetric 2-tensor, in any dimension.

Like the tensor product, the wedge product is defined for two vectors of arbitrary dimension. Notice, too, that the wedge product shares many properties with the cross product. For example, it is easy to verify directly from the definition of the wedge product as the difference of two tensor products obeys the following properties:

$\displaystyle {\bf u} \wedge {\bf u} = 0 ;$
$\displaystyle {\bf v} \wedge {\bf u} = - {\bf u} \wedge {\bf v}$ (anticommutative);
$\displaystyle {\bf u} * \left( {\bf v} \wedge {\bf w} \right) \ne \left( {\bf u} \wedge {\bf v} \right) * {\bf w}^{\mathrm{T}}$ (nonassociative);
$\displaystyle c \left( {\bf u} \wedge {\bf u} \right) = \left( c\,{\bf u} \right) \wedge {\bf u} = {\bf u} \wedge \left( c\,{\bf u} \right) ;$
$\displaystyle {\bf u} \wedge \left( {\bf v} + {\bf w} \right) = {\bf u} \wedge {\bf v} + {\bf u} \wedge {\bf w}$ (distributive);
$\displaystyle {\bf u} * \left( {\bf v} \wedge {\bf w} \right) + {\bf v} * \left( {\bf w} \wedge {\bf u} \right) + {\bf w} * \left( {\bf u} \wedge {\bf v} \right) = 0$ (Jacobi identity).

The wedge product also shares some other important properties with the cross product. The defining characteristics of the cross product are captured by the formulas

$\displaystyle {\bf u} * \left( {\bf u} \wedge {\bf v} \right) * {\bf u}^{\mathrm{T}} = {\bf v} * \left( {\bf u} \wedge {\bf v} \right) * {\bf v}^{\mathrm{T}} ,$
$\displaystyle {\bf u} * \left( {\bf u} \wedge {\bf v} \right) * {\bf v}^{\mathrm{T}} = \left( {\bf u} \cdot {\bf u} \right) \left( {\bf v} \cdot {\bf v} \right) - \left( {\bf u} \cdot {\bf v} \right)^2 = \| {\bf u} \|^2 \, \| {\bf v} \|^2 \sin\theta .$

Moreover, in three dimensions, the entries of the wedge product matrix u ∧ v are, up to sign, the same as the components of the cross product vector u × v.

${\bf u} \wedge {\bf v} = \begin{bmatrix} 0 & u_1 v_2 - u_2 v_1 & u_1 v_3 - u_3 v_1 \\ u_2 v_1 - u_1 v_2 & 0 & u_2 v_3 - u_3 v_2 \\ u_3 v_1 - u_1 v_3 & u_3 v_2 - u_2 v_3 & 0 \end{bmatrix} .$

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An outer product is the tensor product of two coordinate vectors ${\bf u} = \left[ u_1 , u_2 , \ldots , u_m \right]$ and ${\bf v} = \left[ v_1 , v_2 , \ldots , v_n \right] ,$ denoted ${\bf u} \otimes {\bf v} ,$ is an m-by-n matrix W of rank 1 such that its coordinates satisfy $w_{i,j} = u_i v_j .$ The outer product ${\bf u} \otimes {\bf v} ,$ is equivalent to a matrix multiplication ${\bf u} \, {\bf v}^{\ast} ,$ (or ${\bf u} \, {\bf v}^{\mathrm T} ,$ if vectors are real) provided that u is represented as a column $m \times 1$ vector, and v as a column $n \times 1$ vector. Here ${\bf v}^{\ast} = \overline{{\bf v}^{\mathrm T}} .$

Example 4: Let
&omega₁ = 3xdx − 5ydy
and
&omega₂ = 2zdx + 4xdz

Then

$\begin{align*} \omega_1 \wedge \omega_2 &= \left( 3x\,{\text d}x - 5y\,{\text d}y \right) \wedge \left( 2z\,{\text d}x + 4x\,{\text d}z \right) \\ &= 12 x^2 \,{\text d}x \wedge {\text d}z -10yz \,{\text d}y \wedge {\text d}x -20xy \,{\text d}y \wedge {\text d}z \\ &= 12 x^2 \,{\text d}x \wedge {\text d}z +10yz \,{\text d}x \wedge {\text d}y -20xy \,{\text d}y \wedge {\text d}z . \end{align*}$

Here we used identities dx∧dx = 0 and dy∧dx = − dx∧dy.

If &omega₃ = ydz − zdy, then

$\begin{align*} \omega_1 \wedge \omega_2 \wedge \omega_3 &= \left( x^2 \,{\text d}x \wedge {\text d}z +10yz \,{\text d}x \wedge {\text d}y -20xy \,{\text d}y \wedge {\text d}z \right) \wedge \left( y\,{\text d}z - z\,{\text d}y \right) \\ &= - \end{align*}$

Example 4:

The outer product operation can be extended for matrices. If A is an m×n matrix and B is an p×q matrix, then their outer product A⊗B is mp×nq matrix:

${\bf A} \otimes {\bf B} = \begin{bmatrix} a_{1,1} {\bf B} & \cdots & a_{1,n} {\bf B} \\ \vdots & \ddots & \vdots \\ a_{m,1} {\bf B} & \cdots & a_{m,n} {\bf B} \end{bmatrix} .$

Example 5:

Anton, Howard (2005), Elementary Linear Algebra (Applications Version) (9th ed.), Wiley International

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Wedge product