Multilinear maps extend the concept of linear maps to multiple vector spaces. They're crucial in understanding how different vector spaces interact, like in or . 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
framework represents multilinear maps
Constructs single linear map from
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)∗
Multilinear map 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 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 given by {vi⊗wj} where {vi} and {wj} 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)=∑i,jaij(vi⊗wj)
Coefficients in linear combination correspond to multilinear map values
Provides coordinate representation of multilinear map
Example: aij=f(ei,ej) for standard basis vectors ei and ej
Matrix representation of multilinear map reshaped into higher- tensor
Preserves all information about multilinear map
Example: 3D array representation of
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 has dimension dimV⋅dimW⋅dimU
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 induces unique linear map 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) 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
, 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⊗V⊗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⊗V∗⊗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 34=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 {ei⊗ej} where {ei} is basis for V and {ej} is dual basis
Tensor space of type (r,s) isomorphic to multilinear map space
Maps from V∗×⋯×V∗×V×⋯×V to scalar field
Example: (2,1)-tensors isomorphic to trilinear maps 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)-tensor T=∑i,jTjiei⊗ej given by ∑iTii
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
Key Terms to Review (21)
Alternating property: The alternating property refers to a characteristic of certain multilinear maps where the value of the map changes sign when any two of its arguments are swapped. This property is crucial in defining antisymmetric functions and forms, as it helps to ensure that these functions yield zero when any two arguments are equal. This concept is especially important in the study of tensors and multilinear maps, as it highlights the behavior of certain mappings under permutations of their inputs.
Bilinear map: A bilinear map is a function that takes two vectors from two different vector spaces and returns a scalar, satisfying linearity in each argument separately. It is a fundamental concept that connects to the structure of tensor products, as bilinear maps can be used to define tensors. Understanding bilinear maps is essential for exploring how vectors interact in multi-dimensional settings and how these interactions can be captured mathematically.
Contraction: In the context of multilinear maps and tensors, a contraction refers to the process of reducing the order of a tensor by summing over one or more pairs of indices. This operation plays a crucial role in transforming tensors and understanding their properties, especially when dealing with symmetric and alternating tensors, as it allows for the exploration of relationships among various dimensions and simplifies complex expressions.
Contravariant Tensor: A contravariant tensor is a type of tensor that transforms in a specific way under a change of coordinates, specifically by the inverse of the Jacobian matrix associated with the transformation. This means that when you change the basis in a vector space, the components of a contravariant tensor will change according to the inverse of how the basis vectors transform. Contravariant tensors are often associated with vectors and higher-dimensional analogs, reflecting quantities that can be seen as arrows pointing in a certain direction within a coordinate system.
Covariant tensor: A covariant tensor is a mathematical object that transforms according to specific rules when the coordinates of the space are changed. It is characterized by its ability to lower indices and often represents linear functionals that take vectors as inputs and yield scalars, aligning with the properties of multilinear maps and tensors.
Direct Sum: The direct sum is a way to combine two or more subspaces into a new vector space that captures all the elements of the original subspaces without overlap. This concept highlights the idea that if you have two subspaces, their direct sum is made up of all possible sums of vectors from each subspace, ensuring that the intersection of those subspaces contains only the zero vector. This notion is essential for understanding how spaces interact, especially when analyzing their properties, relations to orthogonal complements, and how they can be constructed through tensor products.
Einstein Summation Convention: The Einstein Summation Convention is a notational shorthand used in mathematics and physics, where repeated indices in a term imply summation over those indices. This convention simplifies expressions involving tensors and multilinear maps, allowing for more compact and easier manipulation of complex equations that involve vector and tensor operations.
Engineering: Engineering is the application of mathematical and scientific principles to design, build, and analyze structures, machines, and systems. In the context of multilinear maps and tensors, engineering involves using these mathematical tools to model complex systems in fields such as mechanical, civil, and electrical engineering. Understanding how to work with tensors can lead to improved solutions in real-world engineering problems, from stress analysis in materials to fluid dynamics.
Linear functional: A linear functional is a specific type of linear map that takes a vector from a vector space and returns a scalar, satisfying both linearity properties: additivity and homogeneity. This concept plays a crucial role in understanding how vectors can be transformed into real numbers and connects to the idea of dual spaces, where every vector has an associated linear functional. Additionally, linear functionals help in constructing dual bases that relate back to the original vector space.
Linearity: Linearity refers to a property of mathematical functions or transformations that satisfies two main criteria: additivity and homogeneity. This means that a linear function preserves the operations of addition and scalar multiplication, allowing it to behave predictably under these operations. In various mathematical contexts, such as inner products, multilinear maps, and functional analysis, linearity plays a crucial role in establishing structure and facilitating the understanding of complex systems.
Mixed Tensors: Mixed tensors are mathematical objects that combine multiple types of tensorial components, specifically involving both covariant and contravariant indices. They can be viewed as multilinear maps that take vectors and covectors as inputs and yield a scalar, showcasing the ability to represent relationships between different vector spaces. This blending of different index types allows mixed tensors to play a crucial role in various areas of mathematics, including differential geometry and linear algebra.
Multilinear map: A multilinear map is a function that takes multiple vector arguments and is linear in each argument separately. This means that if you fix all but one argument, the function behaves like a linear transformation in that single argument, allowing it to be expressed as a linear combination of its inputs. These maps are crucial for understanding the relationship between vector spaces and their tensor products, as they help define how these spaces interact and combine.
Order: In the context of multilinear maps and tensors, the term 'order' refers to the number of arguments or inputs that a multilinear map can accept. This concept is fundamental in understanding how tensors function, as the order determines how many vector spaces are involved in the mapping process. Tensors can be visualized as multidimensional arrays, and their order corresponds to the dimensions of these arrays, playing a critical role in defining their properties and operations.
Physics: Physics is the branch of science that deals with the fundamental principles governing matter and energy, encompassing concepts like force, motion, and the interactions between objects. It lays the groundwork for understanding how the universe operates at both macroscopic and microscopic levels, often utilizing mathematical frameworks to describe physical phenomena. The connection between physics and mathematical structures like multilinear maps and tensors becomes essential in modeling complex systems and behaviors in various scientific fields.
Quadratic form: A quadratic form is a homogeneous polynomial of degree two in several variables, typically expressed in the form $Q(x) = x^T A x$, where $x$ is a vector and $A$ is a symmetric matrix. This concept serves as a crucial bridge between linear algebra and geometry, allowing for the analysis of conic sections and providing insight into the properties of matrices and their eigenvalues.
Rank: Rank is a fundamental concept in linear algebra that represents the maximum number of linearly independent column vectors in a matrix. It provides insights into the dimensions of the column space and row space, revealing important information about the solutions of linear systems, the behavior of linear transformations, and the structure of associated tensors.
Schur's Lemma: Schur's Lemma is a fundamental result in representation theory that states that if a linear map between two irreducible representations of a group is invariant under the group action, then this map is either zero or an isomorphism. This lemma connects to multilinear maps and tensors as it provides insight into the structure of these mappings when dealing with representations, particularly in terms of symmetries and invariance.
Tensor notation: Tensor notation is a mathematical language used to describe and manipulate tensors, which are multi-dimensional arrays that generalize scalars, vectors, and matrices. This notation allows for the concise representation of operations involving tensors, facilitating the study of multilinear maps and providing a framework for expressing relationships between different mathematical objects in a structured way.
Tensor product: The tensor product is a mathematical operation that combines two vector spaces to produce a new vector space, which captures multilinear relationships between the original spaces. It is essential for understanding how to create multilinear maps and forms, allowing for the construction of objects that can take multiple vectors and produce scalars. This operation also plays a critical role in defining symmetric and alternating tensors, providing the foundation for analyzing properties related to symmetry and antisymmetry in mathematical objects.
Trilinear map: A trilinear map is a function that takes three vector arguments and is linear in each of those arguments separately. This means that if you hold two arguments fixed and vary the third, the function behaves like a linear transformation with respect to that third argument, and similarly for the other two. Trilinear maps are important as they generalize bilinear maps and relate to the structure of tensors, allowing for operations involving three different vector spaces simultaneously.
Weyl's Theorem: Weyl's Theorem states that for a compact Hermitian operator on a finite-dimensional complex inner product space, the eigenvalues can be organized into a non-decreasing sequence, and the algebraic multiplicity of each eigenvalue equals its geometric multiplicity. This theorem connects with the concepts of multilinear maps and tensors by establishing the foundational properties of operators in vector spaces. It also relates to symmetric and alternating tensors, as these tensors often arise in contexts involving eigenvalues and eigenvectors, showcasing the interplay between different mathematical structures.