Orthogonal group

Lie groups | Quadratic forms | Euclidean symmetries | Linear algebraic groups

In mathematics, the orthogonal group in dimension n, denoted O(n), is the group of distance-preserving transformations of a Euclidean space of dimension n that preserve a fixed point, where the group operation is given by composing transformations. The orthogonal group is sometimes called the general orthogonal group, by analogy with the general linear group. Equivalently, it is the group of n×n orthogonal matrices, where the group operation is given by matrix multiplication (an orthogonal matrix is a real matrix whose inverse equals its transpose). The orthogonal group is an algebraic group and a Lie group. It is compact. The orthogonal group in dimension n has two connected components. The one that contains the identity element is a normal subgroup, called the special orthogonal group, and denoted SO(n). It consists of all orthogonal matrices of determinant 1. This group is also called the rotation group, generalizing the fact that in dimensions 2 and 3, its elements are the usual rotations around a point (in dimension 2) or a line (in dimension 3). In low dimension, these groups have been widely studied, see SO(2), SO(3) and SO(4). The other component consists of all orthogonal matrices of determinant –1. This component does not form a group, as the product of any two of its elements is of determinant 1, and therefore not an element of the component. By extension, for any field F, an n×n matrix with entries in F such that its inverse equals its transpose is called an orthogonal matrix over F. The n×n orthogonalmatrices form a subgroup, denoted O(n, F), of the general linear group GL(n, F); that is More generally, given a non-degenerate symmetric bilinear form or quadratic form on a vector space over a field, the orthogonal group of the form is the group of invertible linear maps that preserve the form. The preceding orthogonal groups are the special case where, on some basis, the bilinear form is the dot product, or, equivalently, the quadratic form is the sum of the square of the coordinates. All orthogonal groups are algebraic groups, since the condition of preserving a form can be expressed as an equality of matrices. (Wikipedia).

Linear Algebra 7.1 Orthogonal Matrices

My notes are available at http://asherbroberts.com/ (so you can write along with me). Elementary Linear Algebra: Applications Version 12th Edition by Howard Anton, Chris Rorres, and Anton Kaul A. Roberts is supported in part by the grants NSF CAREER 1653602 and NSF DMS 2153803.

From playlist Linear Algebra

11H Orthogonal Projection of a Vector

The orthogonal projection of one vector along another.

From playlist Linear Algebra

11J Orthogonal Projection of a Vector

The orthogonal projection of one vector along another.

From playlist Linear Algebra

Orthogonal sets

This is the first video of a linear algebra-series on orthogonality. In this video, I define the notion of orthogonal sets, then show that an orthogonal set without the 0 vector is linearly independent, and finally I show that it's easy to calculate the coordinates of a vector in terms of

From playlist Orthogonality

11I Orthogonal Projection of a Vector

The Orthogonal Projection of one vector along another.

From playlist Linear Algebra

Orthogonal Projections

In this video, I define the concept of orthogonal projection of a vector on a line (and on more general subspaces), derive a very nice formula for it, and show why orthogonal projections are so useful. You might even see the hugging formula again. Enjoy! This is the second part of the ort

From playlist Orthogonality

Symmetric Groups (Abstract Algebra)

Symmetric groups are some of the most essential types of finite groups. A symmetric group is the group of permutations on a set. The group of permutations on a set of n-elements is denoted S_n. Symmetric groups capture the history of abstract algebra, provide a wide range of examples in

From playlist Abstract Algebra

Orthogonal Complements

Orthogonal complements. The direct sum of a subspace and its orthogonal complement. Dimension of the orthogonal complement. The orthogonal complement of the orthogonal complement.

From playlist Linear Algebra Done Right

Orthogonal Matrix Example (Ch5 Pr28)

We look at a rotation matrix as an example of a orthogonal matrix. This is Chapter 5 Problem 28 from the MATH1141/MATH1131 Algebra notes. Presented by Daniel Mansfield from the School of Mathematics and Statistics at UNSW.

From playlist Mathematics 1A (Algebra)

Tobias Braun - Orthogonal Determinants

Basic concepts and notions of orthogonal representations are in- troduced. If X : G → GL(V ) is a K-representation of a nite group G it may happen that its image X(G) xes a non-degenerate quadratic form q on V . In this case X and its character χ : G → K, g 7 → trace(X(g)) are called ortho

From playlist École d'Été 2022 - Cohomology Geometry and Explicit Number Theory

Concentration of Measure on the Compact Classical Matrix Groups - Elizabeth Meckes

Elizabeth Meckes Case Western Reserve Univ May 20, 2014 For more videos, visit http://video.ias.edu

From playlist Mathematics

Abstract Algebra 1.5 : Examples of Groups

In this video, I introduce many important examples of groups. This includes the group of (rigid) motions, orthogonal group, special orthogonal group, the dihedral groups, and the "finite cyclic group" Z/nZ (or Z_n). Email : fematikaqna@gmail.com Code : https://github.com/Fematika/Animatio

From playlist Abstract Algebra

Representation theory: Orthogonality relations

This lecture is about the orthogonality relations of the character table of complex representations of a finite group. We show that these representations are unitary and deduce that they are all sums of irreducible representations. We then prove Schur's lemma describing the dimension of t

From playlist Representation theory

Sylvie PAYCHA - From Complementations on Lattices to Locality

A complementation proves useful to separate divergent terms from convergent terms. Hence the relevance of complementation in the context of renormalisation. The very notion of separation is furthermore related to that of locality. We extend the correspondence between Euclidean structures o

From playlist Algebraic Structures in Perturbative Quantum Field Theory: a conference in honour of Dirk Kreimer's 60th birthday

Marcela Hanzer: Adams’ conjecture on theta correspondence

CIRM VIRTUAL EVENT Recorded during the meeting "Relative Aspects of the Langlands Program, L-Functions and Beyond Endoscopy the May 27, 2021 by the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Jean Petit Find this video and other talks given by worldw

From playlist Virtual Conference

Representation theory: Examples D8, A4, S4, S5, A5

In this talk we calculate the character tables of several small groups: the dihedral group of order 8, and the alternating and symmetric groups on 4 and 5 points. We do this by first finding the 1-dimensional characters, then finding a few other characters by looking at permutation repres

From playlist Representation theory

Algebraic topology: Calculating the fundamental group

This lecture is part of an online course on algebraic topology. We calculate the fundamental group of several spaces, such as a ficure 8, or the complement of a circle in R^3, or the group GL3(R). For the other lectures in the course see https://www.youtube.com/playlist?list=PL8yHsr3EF

From playlist Algebraic topology

RT7.3. Finite Abelian Groups: Convolution

Representation Theory: We define convolution of two functions on L^2(G) and note general properties. Three themes: convolution as an analogue of matrix multiplication, convolution with character as an orthogonal projection on L^2(G), and using using convolution to project onto irreduci

From playlist Representation Theory

Linear Algebra - Lecture 38 - Orthogonal Sets

In this lecture, we discuss orthogonal sets of vectors. We also investigate the idea of an orthogonal basis, as well as orthogonal projections of vectors.

From playlist Linear Algebra Lectures

Steven Bradlow - Exotic components of surface group representation varieties

Steven Bradlow Exotic components of surface group representation varieties, and their Higgs bundle avatars Moduli spaces of Higgs bundles on a Riemann surface correspond to representation varieties for the surface fundamental group. For representations into complex semisimple Lie groups,

From playlist Maryland Analysis and Geometry Atelier

Orthogonal group

Related pages