Children: Linear algebra
The Golden-Thompson inequality is a result in linear algebra and functional analysis, particularly concerning positive semi-definite matrices and traces of exponentials of matrices.
In mathematics, the term "graded" can refer to various concepts depending on the context. Here are a few common interpretations: 1. **Graded Algebra**: In algebra, a graded algebra is an algebraic structure that decomposes into a direct sum of abelian groups (or vector spaces) indexed by non-negative integers. This means that the elements of the algebra can be categorized by their degree, allowing for operations to be defined in a way that respects this grading.
The Gram–Schmidt process is an algorithm used in linear algebra to orthogonalize a set of vectors in an inner product space, most commonly in Euclidean space. The primary goal of this process is to take a finite, linearly independent set of vectors and transform it into an orthogonal (or orthonormal) set of vectors, which are mutually perpendicular to one another or normalized to have unit length.
The Hahn-Banach theorem is a fundamental result in functional analysis, particularly in the study of linear functionals on normed vector spaces. It has several formulations and applications, but its primary statement concerns the extension of linear functionals. ### Statement of the Hahn-Banach Theorem Informally, the theorem asserts that under certain conditions, a bounded linear functional defined on a subspace of a normed vector space can be extended to the whole space without increasing its norm.
Haynsworth's inertia additivity formula provides a way to compute the inertia (the number of positive, negative, and zero eigenvalues) of a block matrix based on the inertia of its individual blocks and their interactions.
Hermite Normal Form (HNF) is a special form of a matrix used in linear algebra, particularly in the context of integer linear algebra. A matrix is in Hermite Normal Form if it satisfies the following conditions: 1. It is an upper triangular matrix: All entries below the main diagonal are zero. 2. The diagonal entries are strictly positive: Each diagonal entry is a positive integer.
A Hilbert space is a fundamental concept in mathematics and quantum mechanics, named after the mathematician David Hilbert. It is a complete inner product space, which is a vector space equipped with an inner product that allows for the measurement of angles and lengths.
The Hilbert–Poincaré series is a concept in algebraic geometry and commutative algebra that links the geometric properties of a variety (or scheme) with algebraic properties of its coordinate ring. Specifically, it provides information about the dimensions of the graded components of this ring.
Homogeneous coordinates are a system of coordinates used in projective geometry, which provides a way to represent points in a projective space. In computer graphics, robotics, and computer vision, homogeneous coordinates are commonly used to simplify various mathematical operations, particularly when dealing with transformations such as translation, rotation, scaling, and perspective projections.
A homogeneous function is a specific type of mathematical function that exhibits a particular property related to scaling.
The Hurwitz determinant is a concept from mathematics, specifically in the area of algebra and stability theory. It is used primarily in the context of systems of differential equations and control theory to analyze the stability of dynamical systems.
A hyperplane is a concept from geometry and linear algebra that refers to a subspace of one dimension less than its ambient space. In simple terms, if you have an \( n \)-dimensional space, a hyperplane would be an \( (n-1) \)-dimensional subspace. Here are some examples to clarify: 1. **In 2D (two-dimensional space)**: A hyperplane is a line. It divides the plane into two halves.
An indeterminate system, also known as an underdetermined system in some contexts, refers to a situation in various fields—such as mathematics, physics, and engineering—where the number of equations is less than the number of unknown variables. This leads to a scenario where there are infinitely many solutions or no solutions at all, depending on the relationships between the equations and the variables. ### In Mathematics: In linear algebra, a system of equations is indeterminate when it has more variables than equations.
Integer points in convex polyhedra refer to the points whose coordinates are integers and that lie within (or on the boundary of) a convex polyhedron defined in a Euclidean space. A convex polyhedron is a three-dimensional geometric figure with flat polygonal faces, straight edges, and vertices, such that a line segment joining any two points in the polyhedron lies entirely inside or on the boundary of the polyhedron.
The International Linear Algebra Society (ILAS) is an organization dedicated to the promotion and advancement of the field of linear algebra and its applications. Founded in 2000, ILAS aims to bring together researchers, educators, and practitioners interested in linear algebra and its numerous applications in various fields such as mathematics, computer science, engineering, and the natural sciences. The society organizes conferences, workshops, and other gatherings to facilitate communication and collaboration among linear algebra researchers.
In functional analysis and operator theory, an **invariant subspace** refers to a subspace of a given vector space that is preserved under the action of a given linear operator. More formally, let \( T: V \to V \) be a linear operator on a vector space \( V \).
Invariants of tensors are scalar quantities derived from the tensor that remain unchanged under certain transformations, typically under coordinate transformations or changes of basis. These invariants are significant in various fields of mathematics, physics, and engineering, notably in the study of material properties in continuum mechanics, the formulation of physical laws, and the analysis of geometric structures. ### Key Concepts: 1. **Tensor Basics**: - Tensors are multi-dimensional arrays that generalize scalars and vectors.
An **invertible matrix** (also known as a non-singular matrix or non-degenerate matrix) is a square matrix \( A \) that has an inverse. This means there exists another matrix \( B \) such that: \[ AB = BA = I \] where \( I \) is the identity matrix of the same dimension as \( A \). A matrix is invertible if and only if its determinant is non-zero (i.e.
The joint spectral radius is a concept from the field of dynamical systems and control theory that deals with the long-term behavior of sets of matrices. It is particularly relevant in the study of systems that can be described by multiple linear transformations, typically when analyzing the stability and robustness of systems involving several processes or state transitions.
Jordan normal form (or Jordan canonical form) is a special form of a square matrix in linear algebra that simplifies the representation of linear transformations. It is particularly useful for studying the properties of linear operators and can be used to perform calculations related to matrix exponentiation, differential equations, and more. A matrix is said to be in Jordan normal form if it is a block diagonal matrix composed of Jordan blocks.