Children: Matrices
Higher-dimensional gamma matrices are generalizations of the familiar Dirac gamma matrices used in quantum field theory, particularly in the context of relativistic quantum mechanics and the formulation of spinors.
Higher spin alternating sign matrices (ASMs) are a generalization of the classical alternating sign matrices, which are combinatorial objects studied in combinatorics and statistical mechanics.
A Hilbert matrix is a specific type of square matrix that is very well-known in numerical analysis and approximation theory.
A **hollow matrix** typically refers to a type of matrix structure where the majority of the elements are zero, and the non-zero elements are arranged in such a way that they form a specific pattern or shape. This term can apply in various mathematical or computational contexts, such as: 1. **Sparse Matrix**: A hollow matrix can be considered a sparse matrix, where most of the elements are zero. Sparse matrices are often encountered in scientific computing, especially when dealing with large datasets.
The Householder transformation is a linear algebra technique used to perform orthogonal transformations of vectors and matrices. It is particularly useful in numerical linear algebra for QR decomposition and in other applications where one needs to reflect a vector across a hyperplane defined by another vector.
A Hurwitz matrix is a specific type of matrix used in the study of stability of systems, particularly in control theory. It is typically associated with determining the stability of a polynomial in one variable. Specifically, a matrix is considered a Hurwitz matrix if all its leading principal minors are positive.
An identity matrix is a special type of square matrix that plays a key role in linear algebra. It is defined as a matrix in which all the elements of the principal diagonal are equal to 1, and all other elements are equal to 0. In mathematical notation, an identity matrix of size \( n \times n \) is denoted as \( I_n \).
An involutory matrix is a square matrix \( A \) that satisfies the property: \[ A^2 = I \] where \( I \) is the identity matrix of the same dimension as \( A \). This means that when the matrix is multiplied by itself, the result is the identity matrix.
An irregular matrix typically refers to a matrix that does not adhere to the standard structure of a regular matrix, which is a rectangular array of numbers with a defined number of rows and columns. Instead, an irregular matrix may have rows of varying lengths, or it may represent a structure where the elements do not conform to a uniform grid.
In mathematics and particularly in linear algebra, a *Jacket matrix* is not a standard term. However, it's possible you may be referring to a *Jacobian matrix*, which is a frequently used concept in differential calculus, especially in the context of multivariable functions. ### Jacobian Matrix The Jacobian matrix describes the rate of change of a vector-valued function with respect to its input vector.
The Jacobian matrix and its determinant play a significant role in multivariable calculus, particularly in the study of transformations and functions of several variables. ### Jacobian Matrix The Jacobian matrix is a matrix of first-order partial derivatives of a vector-valued function.
John Williamson was a British mathematician known for his contributions to the field of mathematics, particularly in the area of algebra and number theory. He was active during the early to mid-20th century and is perhaps best known for his work on matrix theory and quadratic forms. Williamson's most notable contributions include his research on the properties of symmetric matrices and the classification of certain algebraic structures.
Jones calculus is a mathematical framework used in optics to describe the polarization state of light and its transformation through optical devices. It was developed by the physicist R.W. Jones in 1941. This calculus uses a two-dimensional complex vector to represent the state of polarization of light, which can include various types of polarization such as linear, circular, and elliptical.
Krawtchouk matrices are mathematical constructs used in the field of linear algebra, particularly in connection with orthogonal polynomials and combinatorial structures. They arise from the Krawtchouk polynomials, which are orthogonal polynomials associated with the binomial distribution.
An L-matrix generally refers to a specific type of matrix used in the field of mathematics, particularly in linear algebra or optimization. However, the term can vary in meaning depending on the context in which it's used. 1. **Linear Algebra Context:** In linear algebra, an L-matrix might refer to a matrix that is lower triangular, meaning all entries above the diagonal are zero. This is often denoted as \( L \) in contexts such as Cholesky decomposition or LU decomposition.
The Lehmer matrix, named after mathematician D. H. Lehmer, is a specific type of structured matrix that is commonly used in numerical analysis and linear algebra.
A Leslie matrix is a special type of matrix used in demographics and population studies to model the age structure of a population and its growth over time. It is particularly useful for modeling the growth of populations with discrete age classes. The matrix takes into account both the survival rates and birth rates of a population.
Levinson recursion, also known as Levinson-Durbin recursion, is an efficient algorithm used to solve the problem of linear prediction in time series analysis, particularly in the context of autoregressive (AR) modeling. The algorithm is named after the mathematicians Norman Levinson and Richard Durbin, who contributed to its development. The primary goal of Levinson recursion is to recursively compute the coefficients of a linear predictor for a stationary time series, which minimizes the prediction error.
The term "linear group" typically refers to a specific type of group in the context of group theory, a branch of mathematics. Specifically, linear groups are groups of matrices that represent linear transformations in vector spaces. They can be defined over various fields, such as the real numbers, complex numbers, or finite fields.