7.3 Multilinear maps and tensors

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

Top images from around the web for Understanding Multilinear Maps

• 

  Tensor – AnthroWiki View original

  Is this image relevant?

• 

  Linear map - Wikipedia View original

  Is this image relevant?

• 

  linear algebra - How does this Tensor Product basis example is made up? - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  Tensor – AnthroWiki View original

  Is this image relevant?

• 

  Linear map - Wikipedia View original

  Is this image relevant?

1 of 3

Top images from around the web for Understanding Multilinear Maps

• 

  Tensor – AnthroWiki View original

  Is this image relevant?

• 

  Linear map - Wikipedia View original

  Is this image relevant?

• 

  linear algebra - How does this Tensor Product basis example is made up? - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  Tensor – AnthroWiki View original

  Is this image relevant?

• 

  Linear map - Wikipedia View original

  Is this image relevant?

1 of 3

• 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 $(V \otimes W)^*$
• 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 $V \otimes W$ given by $\{v_i \otimes w_j\}$ where $\{v_i\}$ and $\{w_j\}$ are bases for $V$ and $W$
• Multilinear maps expressed as linear combinations of elementary tensors
  • Elementary tensors tensor products of basis vectors
  • Example: bilinear map $f(v,w) = \sum_{i,j} a_{ij} (v_i \otimes w_j)$
• Coefficients in linear combination correspond to multilinear map values
  • Provides coordinate representation of multilinear map
  • Example: $a_{ij} = f(e_i, e_j)$ for standard basis vectors $e_i$ and $e_j$
• 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 $V \times W \to U$ has dimension $\dim V \cdot \dim W \cdot \dim U$

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 $f: V \times W \to U$ induces unique linear map $\tilde{f}: V \otimes W \to U$
• Proof constructs linear map explicitly and demonstrates uniqueness
  • Uses properties of tensor product in construction
  • Example: defining $\tilde{f}(v \otimes w) = f(v,w)$ 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 $(r,s)$ elements of tensor product of $r$ copies of $V$ and $s$ copies of $V^*$
  • Generalizes vectors and linear maps
  • Example: $(2,1)$-tensor element of $V \otimes V \otimes V^*$
• Successive tensor products of $V$ and $V^*$ construct tensor space
  • Order determined by tensor type $(r,s)$
  • Example: space of $(1,2)$-tensors constructed as $V \otimes V^* \otimes V^*$
• Dimension of type $(r,s)$ tensor space $n^{(r+s)}$ for $n$-dimensional $V$
  • Grows rapidly with tensor order
  • Example: space of $(2,2)$-tensors on 3D space has dimension $3^4 = 81$
• Tensor space basis constructed from $V$ basis and $V^*$ dual basis tensor products
  • Generates entire tensor space
  • Example: basis for $(1,1)$-tensors given by $\{e_i \otimes e^j\}$ where $\{e_i\}$ is basis for $V$ and $\{e^j\}$ is dual basis
• Tensor space of type $(r,s)$ isomorphic to multilinear map space
  • Maps from $V^* \times \cdots \times V^* \times V \times \cdots \times V$ to scalar field
  • Example: $(2,1)$-tensors isomorphic to trilinear maps $V^* \times V^* \times V \to \mathbb{F}$

Operations and Applications of Tensor Spaces

• Tensor operations defined through action on tensor product basis
  • Contraction, tensor product, raising/lowering indices
  • Example: contraction of $(1,1)$-tensor $T = \sum_{i,j} T^i_j e_i \otimes e^j$ given by $\sum_i T^i_i$
• 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 $(2,0)$-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 $(2,0)$-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