Multilinear maps extend the concept of linear maps to multiple vector spaces. They're crucial in understanding how different vector spaces interact, like in physics or engineering. Tensors provide a powerful framework for representing these maps.
Tensor products allow us to construct spaces that naturally house multilinear maps. This connection between multilinear algebra and tensor products is key to solving complex problems in fields ranging from quantum mechanics to machine learning.
Multilinear Maps and Tensor Products
Understanding Multilinear Maps
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Multilinear maps generalize bilinear maps to multiple vector spaces
Functions linear in each argument when other arguments are held constant
Example: determinant function for square matrices
Tensor product framework represents multilinear maps
Constructs single linear map from multilinear map
Example: representing bilinear form as matrix
Isomorphism exists between multilinear map space and dual space of tensor product
Facilitates conversion between multilinear maps and tensors
Example: identifying bilinear form with element of ( V ⊗ W ) ∗ (V \otimes W)^* ( V ⊗ W ) ∗
Multilinear map rank relates to corresponding tensor rank
Provides measure of complexity for multilinear maps
Example: rank-one multilinear map corresponds to simple tensor
Composition of multilinear maps with linear maps produces new multilinear maps
Corresponds to operations on tensors in tensor product space
Example: composing bilinear map with linear map yields new bilinear map
Tensor Product Basis and Multilinear Maps
Tensor product basis formed by Kronecker product of input space basis vectors
Generates entire tensor product space
Example: basis for V ⊗ W V \otimes W V ⊗ W given by { v i ⊗ w j } \{v_i \otimes w_j\} { v i ⊗ w j } where { v i } \{v_i\} { v i } and { w j } \{w_j\} { w j } are bases for V V V and W W W
Multilinear maps expressed as linear combinations of elementary tensors
Elementary tensors tensor products of basis vectors
Example: bilinear map f ( v , w ) = ∑ i , j a i j ( v i ⊗ w j ) f(v,w) = \sum_{i,j} a_{ij} (v_i \otimes w_j) f ( v , w ) = ∑ i , j a ij ( v i ⊗ w j )
Coefficients in linear combination correspond to multilinear map values
Provides coordinate representation of multilinear map
Example: a i j = f ( e i , e j ) a_{ij} = f(e_i, e_j) a ij = f ( e i , e j ) for standard basis vectors e i e_i e i and e j e_j e j
Matrix representation of multilinear map reshaped into higher-order tensor
Preserves all information about multilinear map
Example: 3D array representation of trilinear map
Tensor components coefficients in tensor product basis expression
Allows for compact representation of multilinear maps
Example: components of stress tensor in continuum mechanics
Dimension of multilinear map space product of input and output space dimensions
Determines complexity of multilinear map representation
Example: space of bilinear maps V × W → U V \times W \to U V × W → U has dimension dim V ⋅ dim W ⋅ dim U \dim V \cdot \dim W \cdot \dim U dim V ⋅ dim W ⋅ dim U
Tensor Product Basis for Multilinear Maps
Universal Property of Tensor Products
Universal property defines tensor product as "most general" space for multilinear maps
Unique linear map from tensor product space to codomain factors through tensor product
Example: bilinear map f : V × W → U f: V \times W \to U f : V × W → U induces unique linear map f ~ : V ⊗ W → U \tilde{f}: V \otimes W \to U f ~ : V ⊗ W → U
Proof constructs linear map explicitly and demonstrates uniqueness
Uses properties of tensor product in construction
Example: defining f ~ ( v ⊗ w ) = f ( v , w ) \tilde{f}(v \otimes w) = f(v,w) f ~ ( v ⊗ w ) = f ( v , w ) and extending linearly
Tensor product unique up to isomorphism due to universal property
Provides canonical construction for multilinear map spaces
Example: different constructions of tensor product (e.g., algebraic, coordinate-free) yield isomorphic spaces
Reduces multilinear map problems to linear map problems on tensor product spaces
Simplifies analysis and computation
Example: studying properties of multilinear map through associated linear map on tensor product
Facilitates definition of tensor operations based on multilinear map actions
Contraction, outer product defined through universal property
Example: defining tensor contraction as trace of associated linear map
Applications of Universal Property
Tensor product uniqueness enables consistent definitions across contexts
Ensures compatibility of tensor operations in different fields
Example: tensor product in differential geometry consistent with linear algebra definition
Universal property justifies tensor product as natural setting for multilinear algebra
Provides theoretical foundation for tensor methods
Example: use of tensors in general relativity grounded in universal property
Allows for generalization of linear algebra concepts to multilinear setting
Extends notions like rank, trace, and determinant to tensors
Example: defining tensor rank using universal property
Simplifies proofs of tensor product properties
Many results follow directly from universal property
Example: proving associativity of tensor product using universal property
Connects abstract tensor theory with concrete representations
Bridges coordinate-free and component-based approaches
Example: relating abstract tensor product to Kronecker product of matrices
Tensor Spaces from Tensor Products
Constructing Tensor Spaces
Tensors of type ( r , s ) (r,s) ( r , s ) elements of tensor product of r r r copies of V V V and s s s copies of V ∗ V^* V ∗
Generalizes vectors and linear maps
Example: ( 2 , 1 ) (2,1) ( 2 , 1 ) -tensor element of V ⊗ V ⊗ V ∗ V \otimes V \otimes V^* V ⊗ V ⊗ V ∗
Successive tensor products of V V V and V ∗ V^* V ∗ construct tensor space
Order determined by tensor type ( r , s ) (r,s) ( r , s )
Example: space of ( 1 , 2 ) (1,2) ( 1 , 2 ) -tensors constructed as V ⊗ V ∗ ⊗ V ∗ V \otimes V^* \otimes V^* V ⊗ V ∗ ⊗ V ∗
Dimension of type ( r , s ) (r,s) ( r , s ) tensor space n ( r + s ) n^{(r+s)} n ( r + s ) for n n n -dimensional V V V
Grows rapidly with tensor order
Example: space of ( 2 , 2 ) (2,2) ( 2 , 2 ) -tensors on 3D space has dimension 3 4 = 81 3^4 = 81 3 4 = 81
Tensor space basis constructed from V V V basis and V ∗ V^* V ∗ dual basis tensor products
Generates entire tensor space
Example: basis for ( 1 , 1 ) (1,1) ( 1 , 1 ) -tensors given by { e i ⊗ e j } \{e_i \otimes e^j\} { e i ⊗ e j } where { e i } \{e_i\} { e i } is basis for V V V and { e j } \{e^j\} { e j } is dual basis
Tensor space of type ( r , s ) (r,s) ( r , s ) isomorphic to multilinear map space
Maps from V ∗ × ⋯ × V ∗ × V × ⋯ × V V^* \times \cdots \times V^* \times V \times \cdots \times V V ∗ × ⋯ × V ∗ × V × ⋯ × V to scalar field
Example: ( 2 , 1 ) (2,1) ( 2 , 1 ) -tensors isomorphic to trilinear maps V ∗ × V ∗ × V → F V^* \times V^* \times V \to \mathbb{F} V ∗ × V ∗ × V → F
Operations and Applications of Tensor Spaces
Tensor operations defined through action on tensor product basis
Contraction, tensor product, raising/lowering indices
Example: contraction of ( 1 , 1 ) (1,1) ( 1 , 1 ) -tensor T = ∑ i , j T j i e i ⊗ e j T = \sum_{i,j} T^i_j e_i \otimes e^j T = ∑ i , j T j i e i ⊗ e j given by ∑ i T i i \sum_i T^i_i ∑ i T i i
Tensor type concept unifies treatment of geometric and physical quantities
Scalars, vectors, linear transformations all special cases of tensors
Example: stress tensor in continuum mechanics ( 2 , 0 ) (2,0) ( 2 , 0 ) -tensor
Tensor spaces provide framework for multilinear problems in various fields
Physics, engineering, computer science, data analysis
Example: moment of inertia tensor in rigid body dynamics
Coordinate transformations on tensors derived from tensor product structure
Generalizes vector and matrix transformations
Example: transformation law for ( 2 , 0 ) (2,0) ( 2 , 0 ) -tensor under change of basis
Tensor decomposition techniques based on tensor product structure
Singular value decomposition, Tucker decomposition
Example: low-rank approximation of tensors in data compression