Children: Linear algebra
In mathematics, particularly in linear algebra, a conformable matrix refers to matrices that can be operated on together under certain operations, typically matrix addition or multiplication. For two matrices to be conformable for addition, they must have the same dimensions (i.e., the same number of rows and columns). For multiplication, the number of columns in the first matrix must match the number of rows in the second matrix.
The conjugate transpose (also known as the Hermitian transpose) of a matrix is an operation that involves two steps: 1. **Transpose**: First, you transpose the matrix, which means you swap its rows and columns. For example, if you have a matrix \( A \) with elements \( a_{ij} \), its transpose \( A^T \) will have elements \( a_{ji} \).
A constant-recursive sequence is a type of sequence defined by a recurrence relation that is constant in nature, meaning that each term is generated based on a fixed number of previous terms and/or constant values. In other words, the sequence is defined using a recurrence that repeatedly applies the same operation without changing its parameters over time.
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 the context of linear algebra and functional analysis, a **cyclic subspace** is a specific type of subspace generated by the action of a linear operator on a particular vector. Often discussed in relation to operators on Hilbert spaces or finite-dimensional vector spaces, a cyclic subspace can be defined as follows: Let \( A \) be a linear operator on a vector space \( V \), and let \( v \in V \) be a vector.
A defective matrix is a square matrix that does not have a complete set of linearly independent eigenvectors. This means that its algebraic multiplicity (the number of times an eigenvalue occurs as a root of the characteristic polynomial) is greater than its geometric multiplicity (the number of linearly independent eigenvectors associated with that eigenvalue). In other words, a matrix is considered defective if it cannot be diagonalized.
A **definite quadratic form** refers to a specific type of quadratic expression in multiple variables that has particular properties regarding the sign of its output. In mathematical terms, a quadratic form can generally be represented as: \[ Q(\mathbf{x}) = \mathbf{x}^T A \mathbf{x} \] where: - \(\mathbf{x}\) is a vector of variables (e.g., \((x_1, x_2, ...
The Delta operator, often denoted by the symbol \( \Delta \), is a finite difference operator used in mathematics, particularly in the field of difference equations and discrete mathematics.
In mathematics, particularly in the fields of linear algebra and statistics, a dependence relation typically refers to a situation where one variable or set of variables can be expressed as a function of another variable or set of variables. This concept is often contrasted with independence, where variables do not influence each other. ### Linear Algebra: In the context of linear algebra, dependence refers to linear dependence among a set of vectors.
The Dieudonné determinant is a generalized determinant used in the context of matrices over certain fields, particularly in relation to algebraic structures known as noncommutative rings. It arises in the study of the representation theory of groups and certain types of algebras, especially in the context of algebraic groups and linear algebraic groups.
In the context of module theory, which is a branch of abstract algebra, the direct sum of modules is a way to combine two or more modules into a new module.
In linear algebra and functional analysis, the concept of a dual basis is tied to the idea of dual spaces.
In the context of field extensions, the concept of a "dual basis" typically applies within the framework of vector spaces and linear algebra.
The dual norm is a concept from functional analysis, particularly in the context of normed vector spaces. It extends the idea of a norm from a vector space to its dual space, which consists of all continuous linear functionals on that vector space.
In mathematics, particularly in functional analysis and linear algebra, the concept of the **dual space** is important in studying vector spaces and linear maps. ### Definition Given a vector space \( V \) over a field \( F \) (commonly the real numbers \( \mathbb{R} \) or complex numbers \( \mathbb{C} \)), the **dual space** \( V^* \) is defined as the set of all linear functionals on \( V \).
The term "eigenoperator" is generally used in the context of quantum mechanics or linear algebra, where it is analogous to the concept of an "eigenvalue" and "eigenvector." In these fields, an operator is a mathematical object that acts on elements in a given space (like a vector space). An eigenoperator can be thought of as a particular kind of operator that has a specific eigenstate (or eigenvector) associated with it.
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.