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Mathematics for Physics

156 Alternating k-linear function

A multilinear mapping of several vectors that changes sign when two inputs are swapped that is used to form determinants and alternating tensors.

definition (Alternating k-linear function) A multilinear function from the \(k\)-fold product of a vector space to the real numbers, \(f: V^k \rightarrow \mathbb{R}\), where the following conditions apply:

  • \(f(v_{\sigma(1)}, \dots, v_{\sigma(k)}) = (\text{sgn } \sigma) f(v_1, \dots, v_k)\) for all \(\sigma \in S_k\).
  • \(f(v_1, \dots, v_i, \dots, v_j, \dots, v_k) = -f(v_1, \dots, v_j, \dots, v_i, \dots, v_k)\) for any interchange of two arguments.
  • \(f(v_1, \dots, v_k) = 0\) whenever two of the vectors \(v_1, \dots, v_k\) are equal.

where

  • \(V\) is a vector space.
  • \(V^k\) is the \(k\)-fold Cartesian product \(V \times \dots \times V\).
  • \(k\) is a positive integer representing the degree of the function.
  • \(S_k\) is the permutation group of \(k\) objects.
  • \(\text{sgn } \sigma\) is the sign of the permutation \(\sigma\).
  • \(v_1, \dots, v_k \in V\) are vectors.
  • \(A^k(V)\) is the vector space of all alternating \(k\)-linear functions on \(V\).

Note:

  • \(A^k(V)\) is also written \(\Lambda^k(V^*)\).
  • k-covector, multicovector of degree k, and alternating k-tensor are synonyms for an alternating \(k\)-linear function.

156.1 Elementary Example

156.1.1 Simple

An alternating \(2\)-linear map on three labeled vectors changes sign under a swap and vanishes on a repeated pair.

\[ f : V \times V \rightarrow \mathbb{R} \]

\[ V = \{ e_{1},\ e_{2},\ e_{3} \} \]

\[ f(e_{1},e_{2}) = 1,\quad f(e_{2},e_{1}) = -1,\quad f(e_{1},e_{1}) = 0 \]

where

  • \(f\) is an alternating \(2\)-linear function.
  • \(k = 2\) is the number of vector inputs.
  • \(V\) is the set of input basis vectors.

156.1.2 General

For \(k = 3\), an alternating \(3\)-linear map on \(\mathbb{R}^{3}\) is the determinant of the matrix with those three vectors as columns.

\[ f : \mathbb{R}^{3} \times \mathbb{R}^{3} \times \mathbb{R}^{3} \rightarrow \mathbb{R} \]

\[ f(u,v,w) = \det\begin{pmatrix} u^{1} & v^{1} & w^{1} \\ u^{2} & v^{2} & w^{2} \\ u^{3} & v^{3} & w^{3} \end{pmatrix} \]

\[ f(e_{1},e_{2},e_{3}) = 1,\quad f(e_{2},e_{1},e_{3}) = -1 \]

where

  • \(f\) is an alternating \(3\)-linear function.
  • \(e_{1}, e_{2}, e_{3}\) are the standard basis vectors of \(\mathbb{R}^{3}\).