Representation theory of groups

Matrix coefficient

In mathematics, a matrix coefficient (or matrix element) is a function on a group of a special form, which depends on a linear representation of the group and additional data. Precisely, it is a function on a compact topological group G obtained by composing a representation of G on a vector space V with a linear map from the endomorphisms of V into V 's underlying field. It is also called a representative function. They arise naturally from finite-dimensional representations of G as the matrix-entry functions of the corresponding matrix representations. The Peter–Weyl theorem says that the matrix coefficients on G are dense in the Hilbert space of square-integrable functions on G. Matrix coefficients of representations of Lie groups turned out to be intimately related with the theory of special functions, providing a unifying approach to large parts of this theory. Growth properties of matrix coefficients play a key role in the classification of irreducible representations of locally compact groups, in particular, reductive real and p-adic groups. The formalism of matrix coefficients leads to a generalization of the notion of a modular form. In a different direction, mixing properties of certain dynamical systems are controlled by the properties of suitable matrix coefficients. (Wikipedia).

Video thumbnail

What is a Matrix?

What is a matrix? Free ebook http://tinyurl.com/EngMathYT

From playlist Intro to Matrices

Video thumbnail

Linear Algebra for Computer Scientists. 12. Introducing the Matrix

This computer science video is one of a series of lessons about linear algebra for computer scientists. This video introduces the concept of a matrix. A matrix is a rectangular or square, two dimensional array of numbers, symbols, or expressions. A matrix is also classed a second order

From playlist Linear Algebra for Computer Scientists

Video thumbnail

Understanding Matrices and Matrix Notation

In order to do linear algebra, we will have to know how to use matrices. So what's a matrix? It's just an array of numbers listed in a grid of particular dimensions that can represent the coefficients and constants from a system of linear equations. They're fun, I promise! Let's just start

From playlist Mathematics (All Of It)

Video thumbnail

Ex: Write a System of Equations as a Matrix Equation (3x3)

This video explains how to write a matrix equation for a system of three equations with three unknowns. http://mathispower4u.com

From playlist Matrix Equations

Video thumbnail

Using a Matrix Equation to Solve a System of Equations

This video shows how to solve a system of equations by using a matrix equation. The graphing calculator is integrated into the lesson. http://mathispower4u.yolasite.com/ http://mathispower4u.wordpress.com/

From playlist Matrix Equations

Video thumbnail

Find the Coefficient Matrix of a System to Determine Eigenvalues of a 2 by 2 Matrix

This video explains how to determine the coeifficient matrix for the system of equations to find the eigenvalues of a 2 by 2 matrix.

From playlist Eigenvalues and Eigenvectors

Video thumbnail

Matrix Addition, Subtraction, and Scalar Multiplication

This video shows how to add, subtract and perform scalar multiplication with matrices. http://mathispower4u.yolasite.com/ http://mathispower4u.wordpress.com/

From playlist Introduction to Matrices and Matrix Operations

Video thumbnail

Homogeneous Systems: Given a Coefficient Matrix, Solve Ax=0

This video explains how to solve the vector equation Ax=0 given a coefficient matrix.

From playlist Rank and Homogeneous Systems

Video thumbnail

Matrices | Determinant of a Matrix (Part 1) | Don't Memorise

What is the Determinant of a Matrix? Is it unique? What are its applications? In this video, we will learn: 0:00 symbol of the determinant of a matrix 0:34 applications of the determinant of a matrix 0:59 only square matrices have determinant Watch Determinant of a Matrix (Part 2) here

From playlist Matrices

Video thumbnail

RT8.3. Finite Groups: Projection to Irreducibles

Representation Theory: Having classified irreducibles in terms of characters, we adapt the methods of the finite abelian case to define projection operators onto irreducible types. Techniques include convolution and weighted averages of representations. At the end, we state and prove th

From playlist Representation Theory

Video thumbnail

Solve a System of Linear Equations Using Cramer's Rule (4 by 4)

This video explains how to solve a system of 4 equations with 4 unknowns using Cramer's Rule.

From playlist The Determinant of a Matrix

Video thumbnail

Fan Yang (U Penn) -- Sample canonical correlation coefficients of high-dimensional random vectors

We study the the sample correlation between two ensembles of high dimensional random vectors from the perspective of canonical-correlation analysis (CCA). Assuming almost sharp moment assumptions on the vector entries, we prove that the finite rank correlations will lead to outliers in the

From playlist Northeastern Probability Seminar 2020

Video thumbnail

55 - Matrix representation of linear maps (continued)

Algebra 1M - international Course no. 104016 Dr. Aviv Censor Technion - International school of engineering

From playlist Algebra 1M

Video thumbnail

Lecture 5 (CEM) -- TMM Using Scattering Matrices

This lecture formulates a stable transfer matrix method based on scattering matrices. The scattering matrices adopted here are greatly improved from the literature and are consistent with convention. The lecture ends with some advanced topics like dispersion analysis, cascading and doubl

From playlist UT El Paso: CEM Lectures | CosmoLearning.org Electrical Engineering

Video thumbnail

Lec 11 | MIT RES.6-008 Digital Signal Processing, 1975

Lecture 11: Representation of linear digital networks Instructor: Alan V. Oppenheim View the complete course: http://ocw.mit.edu/RES6-008S11 License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu

From playlist MIT RES.6-008 Digital Signal Processing, 1975

Video thumbnail

PCA, SVD

Linear dimensionality reduction: principal components analysis (PCA) and the singular value decomposition (SVD)

From playlist cs273a

Video thumbnail

Samit Dasgupta: An introduction to to auxiliary polynomials in transcendence theory, Lecture III

Broadly speaking, transcendence theory is the study of the rationality or algebraicity properties of quantities of arithmetic or analytic interest. For example, Hilbert’s 7th problem asked ”Is a b always transcendental if a 6= 0, 1 is algebraic and b is irrational algebraic?” An affirmativ

From playlist Harmonic Analysis and Analytic Number Theory

Video thumbnail

Math 060 Linear Algebra 12 100614: Change of Basis, ct'd.

Change of bases in abstract vector spaces; transition matrix from one basis to another; example of determining and using the transition matrix; transition matrices are invertible; invertible matrices can be realized as transition matrices

From playlist Course 4: Linear Algebra

Video thumbnail

Math 060 100617C Change of Basis

Change of basis in abstract vector spaces - finding the transition matrix from one coordinate system to another. Motivation: what does the transition matrix really mean (in the case of Euclidean space)? The transition matrix in the case of general vector spaces. Why does it work? Obser

From playlist Course 4: Linear Algebra (Fall 2017)

Video thumbnail

Ex 2: Determinant of 3x3 Matrix - Diagonal Method

This video provides an example of how to calculate the determinant using the diagonal method. Site: http://mathispower4u.com

From playlist The Determinant of a Matrix

Related pages

Schur orthogonality relations | Lie group | Character theory | Vector space | Israel Gelfand | Dynamical system | Riesz representation theorem | Élie Cartan | Group (mathematics) | Dual basis | Hypergeometric function | Modular form | Peter–Weyl theorem | Special functions | Linear map | Jacobi polynomials | Mathematics | Field (mathematics) | Global field | Standard basis | Algebraic geometry | Langlands program | Mixing (mathematics) | Number theory | Compact group | Real analytic Eisenstein series | Legendre polynomials | Trigonometric functions | Hilbert space | Theta function | Function composition | Issai Schur | Orthogonal polynomials | Matrix (mathematics) | Endomorphism | Locally compact group