155 Alternating Function
A property of a function under which swapping two inputs multiplies the value by minus one that is used to build determinants of matrices.
definition (Alternating Function) A property of a \(k\)-linear function from the \(k\)-fold product of a vector space to the real numbers, \(T: V^k \rightarrow \mathbb{R}\), where the following conditions apply:
- \(T(v_{\sigma(1)}, \dots, v_{\sigma(k)}) = (\text{sgn } \sigma) T(v_1, \dots, v_k)\) for every permutation \(\sigma \in S_k\).
- \(T(v_1, \dots, v_i, \dots, v_j, \dots, v_k) = -T(v_1, \dots, v_j, \dots, v_i, \dots, v_k)\) for any interchange of two arguments.
- \(T(v_1, \dots, v_k) = 0\) 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 \(V \times \dots \times V\).
- \(S_k\) is the permutation group of \(k\) objects.
- \(\text{sgn } \sigma\) is the sign of the permutation \(\sigma\), which is \(+1\) if the permutation is even and \(-1\) if it is odd.
- \(v_1, \dots, v_k\) are vectors in \(V\).
- \(k\) is a positive integer representing the degree of the function.
155.1 Elementary Example
155.1.1 Simple
An alternating map on two vectors changes sign under a swap and is zero when the two inputs are equal.
\[ T : V \times V \rightarrow \mathbb{R} \]
\[ V = \{ e_{1},\ e_{2} \} \]
\[ T(e_{1},e_{2}) = 1,\quad T(e_{2},e_{1}) = -1,\quad T(e_{1},e_{1}) = 0,\quad T(e_{2},e_{2}) = 0 \]
where
- \(T\) is an alternating bilinear map.
- \(V\) is the set of two basis vectors used as inputs.
- \(e_{1}, e_{2}\) are those basis vectors.
155.1.2 General
In three dimensions the same rule is the determinant pairing of two vectors in \(\mathbb{R}^{3}\) via the standard cross-product identity in the plane of the first two axes.
\[ T : \mathbb{R}^{3} \times \mathbb{R}^{3} \rightarrow \mathbb{R} \]
\[ T(u,v) = u^{1} v^{2} - u^{2} v^{1} \]
\[ T(e_{1},e_{2}) = 1,\quad T(e_{2},e_{1}) = -1,\quad T(e_{1},e_{1}) = 0 \]
where
- \(u = (u^{1},u^{2},u^{3})\) and \(v = (v^{1},v^{2},v^{3})\) are vectors in \(\mathbb{R}^{3}\).
- \(e_{1}, e_{2}, e_{3}\) are the standard basis vectors of \(\mathbb{R}^{3}\).
155.2 References
- Lovett, S. Differential Geometry of Manifolds. — alternating: sign change under swap; zero when two arguments are equal.
- Alternating Function
- Alternating k-linear function
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