Representation theory of groups
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).
What is a matrix? Free ebook http://tinyurl.com/EngMathYT
From playlist Intro to Matrices
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
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)
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
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
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
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
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
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
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
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
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
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Algebra 1M - international Course no. 104016 Dr. Aviv Censor Technion - International school of engineering
From playlist Algebra 1M
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
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
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
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
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)
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