Multilinear algebra

Multilinear algebra is a branch of mathematics that extends linear algebra by dealing with multilinear functions, which are functions that are linear in each of several arguments. This area of study is essential for understanding vector spaces and can be thought of as a natural progression from linear algebra into more complex structures.

Clifford algebras are a type of algebra associated with a quadratic form on a vector space. They arise in various areas of mathematics and physics, particularly in geometry, algebra, and the theory of spinors. The concept was introduced by the mathematician William Kingdon Clifford in the late 19th century.

Invariant theory is a branch of mathematics, particularly in the fields of algebra, geometry, and representation theory, that studies properties of mathematical objects that remain unchanged (or invariant) under transformations from a certain group. The most common transformations considered are linear transformations, but the theory can also apply to more general transformation groups. Historically, invariant theory originated in the 19th century, with significant contributions from mathematicians such as David Hilbert and Hermann Weyl.

Monoidal categories are a fundamental concept in category theory, providing a framework that captures notions of multiplicative structures in a categorical setting. A monoidal category consists of a category equipped with a tensor product (which can be thought of as a kind of "multiplication" between objects), an identity object, and certain coherence conditions that ensure the structure behaves well.

Tensors are mathematical objects that generalize scalars, vectors, and matrices to higher dimensions. They are fundamental in various fields, including physics, engineering, and machine learning, particularly in deep learning. Here’s a brief overview of what tensors are: 1. **Definition**: A tensor is essentially a multi-dimensional array that can be used to represent data. Tensors can have any number of dimensions. - A **scalar** (a single number) is a 0-dimensional tensor.

An alternating multilinear map is a special type of multilinear function that takes several input arguments from a vector space and has the property of being alternating. Here's a more detailed breakdown of what this means: 1. **Multilinear Map**: A function \( f: V_1 \times V_2 \times \ldots \times V_n \to W \) is called multilinear if it is linear in each of its \( n \) arguments.

The Berezin integral is a concept from mathematical physics, particularly in the field of supersymmetry and quantum mechanics. It is a type of integral used for integrating functions of Grassmann variables, which are anticommuting variables that the Berezin calculus is built around. Grassmann variables play a fundamental role in the formulation of supersymmetry, where they are used to describe fermionic fields or states.

A bilinear map is a mathematical function defined on two vector spaces (or modules) that is linear in each of its arguments when the other is held fixed.

The Binet–Cauchy identity is a result in combinatorics and linear algebra that relates the determinants of matrices and their block structures. It provides a way to compute the determinant of a block matrix in terms of the determinants of its components.

A bivector is a geometric object that arises in the context of multivector algebra, particularly within the framework of geometric algebra. It represents an oriented area element and can be thought of as being associated with the plane spanned by two vectors in a vector space. In more formal terms, a bivector is defined as the exterior (or wedge) product of two vectors.

Cryptographic multilinear maps are advanced mathematical constructs used in cryptography, particularly in advanced protocols and schemes such as functional encryption, attribute-based encryption, and other complex cryptographic primitives. They generalize linear maps (or bilinear maps) to higher dimensions, allowing for operations involving multiple inputs that can be combined in sophisticated ways.

Cubic form typically refers to the mathematical representation of a cubic equation or polynomial, which is a polynomial of degree three.

Discrete exterior calculus (DEC) is a mathematical framework that extends concepts from traditional differential geometry and exterior calculus to discrete settings. It is particularly useful in computational applications, especially in numerical simulations, computer graphics, and finite element methods. In traditional exterior calculus, one deals with smooth manifolds and differential forms, which allow the formulation of concepts like integration over manifolds, differential operators like the exterior derivative, and cohomology.

Einstein notation, also known as Einstein summation convention, is a notational scheme used primarily in the fields of mathematics and physics to simplify expressions involving tensors and multi-indexed arrays. It was introduced by the physicist Albert Einstein in the context of his work on the theory of relativity. The key principle of Einstein notation is that when an index variable appears twice in a single term, it implies a summation over that index.

Exterior algebra is a mathematical framework used primarily in the fields of linear algebra, differential geometry, and algebraic topology. It provides a way to construct and manipulate multi-linear forms and generalized notions of vectors in a vector space. The key components of exterior algebra are: 1. **Vector Spaces**: Exterior algebra begins with a vector space \( V \) over a field (usually the real or complex numbers).

Gama's Theorem, often spelled as Gamas Theorem, is a concept in the field of computational geometry, particularly related to the study of convex polytopes and their properties. It states that in a convex polytope, the number of facets (or faces) of a particular dimension is related to the vertices and edges of the polytope, following certain combinatorial relationships.

A glossary of tensor theory typically includes definitions and explanations of key terms and concepts related to tensors and their applications in fields such as mathematics, physics, and engineering. Here are some important terms that are often included: ### A - **Alignment**: The relationship between two tensors that involve certain conditions of their components in relation to each other and the coordinate systems used.

The HOSVD (Higher-Order Singular Value Decomposition) is a mathematical tool used in tensor decomposition, which is particularly useful in the fields of control theory, signal processing, and machine learning for tasks involving multi-way data or tensor representations. In the context of Tensor Product (TP) functions and quasi-linear parameter-varying (qLPV) models, the HOSVD can be applied to represent these complex systems in a more compact and interpretable form.

Higher-order singular value decomposition (HOSVD) is an extension of the traditional singular value decomposition (SVD) to tensor data, which are multi-dimensional generalizations of matrices. While a matrix is a two-dimensional array (with rows and columns), a tensor can have three or more dimensions, commonly referred to as modes.

A **homogeneous polynomial** is a polynomial whose terms all have the same total degree. In more formal terms, a polynomial \( P(x_1, x_2, \ldots, x_n) \) is called homogeneous of degree \( d \) if every term in the polynomial is of degree \( d \).

The hyperdeterminant is a generalization of the determinant concept for multi-dimensional arrays, or tensors. While a determinant applies to square matrices (two-dimensional arrays), the hyperdeterminant extends this idea to higher-dimensional arrays, specifically to tensors of order \( n \).

The interior product, also known as the inner product or dot product, is a mathematical operation that takes two vectors and produces a scalar. It is a fundamental concept in linear algebra and has applications in various fields, including physics, engineering, and computer science.

Lagrange's identity is a mathematical formula that relates the sums of squares of two sets of variables. It is often stated in the context of inner product spaces or in terms of quadratic forms.

A **multilinear map** is a type of mathematical function that takes multiple vector inputs and is linear in each of its arguments.

Multilinear multiplication refers to a mathematical operation involving multiple variables or tensors, where the product is linear in each argument separately. In the context of tensors, it involves evaluating products in a way that maintains linearity with respect to each of the involved tensors. ### Key Concepts: 1. **Multilinearity**: A function is multilinear if it is linear in each of its arguments independently.

Multilinear subspace learning refers to a set of techniques in machine learning and statistics used to analyze and represent data that exists in a multi-dimensional space. While traditional linear subspace methods (like Principal Component Analysis, PCA) focus on linear relationships within data, multilinear methods extend these concepts to accommodate data that can be best modeled in a higher-dimensional space with multiple modes or tensor structures.

A multivector is an algebraic concept used primarily in the context of geometric algebra and vector calculus. It extends the idea of scalars (0D), vectors (1D), and bivectors (2D) to higher dimensions, providing a unified framework for various mathematical objects. In more detail: 1. **Definition**: A multivector is an element of a geometric algebra that can be expressed as a linear combination of scalars, vectors, bivectors, and higher-dimensional entities.

A **paravector** is a mathematical concept used in the context of geometric algebra and Clifford algebra. Specifically, it refers to an extension of the traditional vector space concepts by incorporating additional types of elements, such as bivectors and higher-dimensional geometric entities.

The Pfaffian is a mathematical construct associated with skew-symmetric matrices, which are square matrices \(A\) satisfying the property \(A^T = -A\). The Pfaffian provides a scalar value that can be thought of as a sort of "square root" of the determinant for skew-symmetric matrices.

Plücker coordinates are a system of homogeneous coordinates used to represent lines in projective space, particularly in three-dimensional projective space \( \mathbb{P}^3 \). They are named after the mathematician Julius Plücker.

Skew lines are lines that do not intersect and are not parallel. They exist in three-dimensional space. Unlike parallel lines, which are always the same distance apart and will never meet, skew lines are positioned such that they are not on the same plane. Consequently, they cannot intersect. For example, consider two lines in a room: one line lying along the edge of a table and another line running across the ceiling.

Symmetric algebra is a fundamental construction in algebra, particularly in the context of algebraic geometry and commutative algebra. Specifically, it is associated with the idea of forming polynomials from elements of a vector space or an algebra.

Tensor algebra is a mathematical framework that extends the concepts of linear algebra to accommodate tensors, which are multi-dimensional arrays that generalize scalars, vectors, and matrices. In simpler terms, tensors can represent data in more complex ways compared to traditional linear algebra structures. ### Key Concepts in Tensor Algebra: 1. **Tensors**: - A scalar is a 0th-order tensor. - A vector is a 1st-order tensor.

A tensor field is a mathematical construct that generalizes the concept of scalars and vectors to higher dimensions, allowing for the representation of more complex relationships in a variety of contexts, particularly in physics and engineering. ### Definition **Tensor**: A tensor is a multi-dimensional array of numerical values that transforms according to specific rules under a change of coordinates. Tensors can be classified based on their rank (or order): - **Scalar**: A tensor of rank 0 (single number).

The tensor product of algebras is a construction in the field of mathematics that allows for the combination of algebraic structures, specifically algebras over a field. It takes two algebras and creates a new algebra which captures the information of both original algebras in a way that respects their algebraic operations. Here's a more detailed breakdown: ### Definitions 1.

Tensor rank decomposition is a mathematical concept used to express a tensor as a sum of simpler tensors, often referred to as "rank-one tensors." Tensors can be thought of as multi-dimensional arrays, and they generalize matrices (which are two-dimensional tensors) to higher dimensions.

Tensor reshaping is the process of changing the shape or dimensions of a tensor without altering its data. A tensor is a mathematical object that can be thought of as a generalization of scalars, vectors, and matrices to higher dimensions. In machine learning and data manipulation, tensors are commonly used to represent multi-dimensional data.