The characteristic polynomial is a polynomial that is derived from a square matrix and is used in linear algebra to provide important information about the matrix, particularly its eigenvalues.

A glossary of linear algebra typically includes key terms and concepts that are fundamental to the study and application of linear algebra. Here’s a list of some important terms you might find in such a glossary: ### Glossary of Linear Algebra 1. **Vector**: An element of a vector space; often represented as a column or row of numbers. 2. **Matrix**: A rectangular array of numbers arranged in rows and columns.

A **controlled invariant subspace** is a concept from control theory and linear algebra that pertains to the behavior of dynamical systems. In the context of linear systems, it often refers to subspaces of the state space that are invariant under the action of the system's dynamics when a control input is applied.

A **convex cone** is a fundamental concept in mathematics, particularly in linear algebra and convex analysis.

In linear algebra and functional analysis, the concept of a dual basis is tied to the idea of dual spaces.

Eigenplane is a technique related to the fields of machine learning and computer vision that typically involves dimensionality reduction and representation learning. It is often used to represent complex data by finding a lower-dimensional space that captures the essential features of the data while retaining its important characteristics.

A finite von Neumann algebra is a special type of von Neumann algebra that satisfies certain properties related to its structure and its trace. Von Neumann algebras are a class of *-algebras of bounded operators on a Hilbert space that are closed in the weak operator topology. They play a central role in functional analysis and quantum mechanics.

In linear algebra, a **frame** is a concept that generalizes the idea of a basis in a vector space. While a basis is a set of linearly independent vectors that spans the vector space, a frame can include vectors that are not necessarily independent and may provide redundancy. This redundancy is beneficial in various applications, particularly in signal processing and data analysis.

Loewner order, named after the mathematician Charles Loewner, is a way to compare positive definite matrices. In particular, for two symmetric matrices \( A \) and \( B \), we say that \( A \) is less than or equal to \( B \) in the Loewner order, denoted \( A \preceq B \), if the matrix \( B - A \) is positive semidefinite.

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.

Leibniz's formula for the determinant of an \( n \times n \) matrix provides a way to compute the determinant based on permutations of the matrix indices.

In mathematics, particularly in the context of linear algebra and functional analysis, a **linear form** (or linear functional) is a specific type of function that satisfies certain properties. Here are the main characteristics: 1. **Linear Transformation**: A linear form maps a vector from a vector space to a scalar.

In linear algebra, the nonnegative rank of a matrix is a measure of the smallest number of nonnegative rank-one matrices that can be summed to produce the original matrix. A rank-one matrix can be expressed as the outer product of two vectors.

A **quasinorm** is a generalization of the concept of a norm used in mathematical analysis, particularly in functional analysis and vector spaces. While a norm is a function that assigns a non-negative length or size to vectors (satisfying certain properties), a quasinorm relaxes some of these requirements.

The Restricted Isometry Property (RIP) is a concept from the field of compressed sensing and high-dimensional geometry. It describes a condition under which a linear transformation approximately preserves the distances between a limited number of vectors in a high-dimensional space.

A matrix difference equation is a mathematical equation that describes the relationship between a sequence of vectors or matrices at discrete time intervals. Specifically, it generalizes the concept of a scalar difference equation to the context of matrices or vectors.

The orientation of a vector bundle is a concept from differential geometry and algebraic topology that is related to the notion of orientability of the fibers of the bundle. A vector bundle \( E \) over a topological space \( X \) consists of a base space \( X \) and, for each point \( x \in X \), a vector space \( E_x \) attached to that point. The vector spaces are called the fibers of the bundle. ### Definition of Orientation 1.

The LINEAR (Lincoln Near-Earth Asteroid Research) project was a program designed to detect and track near-Earth objects, including asteroids and comets. Established in 1998, LINEAR made significant contributions to the discovery of various celestial objects.

The term "pairing" can refer to different concepts depending on the context. Here are a few common interpretations: 1. **Cooking and Beverages**: In culinary contexts, pairing often refers to the art of matching foods with beverages (like wine or beer) to enhance the overall dining experience. For example, red wine is commonly paired with red meat, while white wine is often paired with seafood.