182 Multilinear Function
A mapping of several vectors that is linear in each vector variable separately that is used to define tensors.
definition (Multilinear Function) A 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 for each argument \(i\) (\(1 \leq i \leq k\)):
- \(T(v_1, \dots, v_i + v_i', \dots, v_k) = T(v_1, \dots, v_i, \dots, v_k) + T(v_1, \dots, v_i', \dots, v_k)\)
- \(T(v_1, \dots, av_i, \dots, v_k) = a \cdot T(v_1, \dots, v_i, \dots, v_k)\)
where
- \(V\) is a vector space over \(\mathbb{R}\).
- \(V^k\) is the \(k\)-fold Cartesian product \(V \times \dots \times V\).
- \(k\) is a positive integer representing the number of arguments, also called the degree of the function.
- \(v_1, \dots, v_k, v_i' \in V\) are vectors.
- \(a \in \mathbb{R}\) is a scalar.
182.1 Elementary Example
182.1.1 Simple
A bilinear map is linear in each slot separately. Values on two basis vectors need not change sign under a swap.
\[ T : V \times V \rightarrow \mathbb{R} \]
\[ V = \{ e_{1},\ e_{2},\ e_{3} \} \]
\[ T(e_{1},e_{2}) = 5,\quad T(e_{2},e_{1}) = 3,\quad T(e_{1},e_{1}) = 2 \]
where
- \(T\) is a multilinear function with two inputs.
182.1.2 General
On \(\mathbb{R}^{3}\), a bilinear map is a general \(3 \times 3\) matrix of components.
\[ (T_{ij}) = \begin{pmatrix} 2 & 5 & 0 \\ 3 & 1 & 4 \\ 0 & 7 & 6 \end{pmatrix} \]
\[ T(u,v) = \sum_{i,j} T_{ij}\, u^{i}\, v^{j} \]
where
- \(T_{ij}\) are the components of the bilinear map \(T\).
- Alternating Function
- Alternating k-linear function
- Boundary Operation
- Bundle
- Cartesian Product
- Constant Metric Tensor
- Contravariant Rank
- Covariant Rank
- Covariant, Contravariant, and Coordinate Systems
- Differential Forms
- Differential k-Form
- Edge of a Domain
- Exterior Algebra and Exterior Derivatives
- Forms
- Functions
- Inverse Metric
- k-Covector
- k-Covector Field
- k-Fold Product
- k-form
- k-Tensor
- Lie
- Lie Algebra
- Line Element
- Linear Functional
- Lorentzian Manifold
- Metric Tensor
- Multilinear Function
- Nondegenerate
- Nondegenerate Bilinear Form
- One-Form
- Smooth
- Smooth Assignment
- Smooth Mapping
- Stoke’s Theorem and the Fundamental Theorem of Calculus
- Symmetric Array
- Tangent Space
- Tensor Field
- Tensors
- Two-Form
- Type-(0,2) Tensor Field
- Type-(q,r) Tensor