In mathematics, the indefinite orthogonal group, O(p, q) is the Lie group of all linear transformations of an n-dimensional real vector space that leave invariant a nondegenerate, symmetric bilinear form of signature (p, q), where n = p + q. It is also called the pseudo-orthogonal group or generalized orthogonal group. The dimension of the group is n(n − 1)/2. The indefinite special orthogonal group, SO(p, q) is the subgroup of O(p, q) consisting of all elements with determinant 1. Unlike in the definite case, SO(p, q) is not connected – it has 2 components – and there are two additional finite index subgroups, namely the connected SO+(p, q) and O+(p, q), which has 2 components – see for definition and discussion. The signature of the form determines the group up to isomorphism; interchanging p with q amounts to replacing the metric by its negative, and so gives the same group. If either p or q equals zero, then the group is isomorphic to the ordinary orthogonal group O(n). We assume in what follows that both p and q are positive. The group O(p, q) is defined for vector spaces over the reals. For complex spaces, all groups O(p, q; C) are isomorphic to the usual orthogonal group O(p + q; C), since the transform changes the signature of a form. This should not be confused with the indefinite unitary group U(p, q) which preserves a sesquilinear form of signature (p, q). In even dimension n = 2p, O(p, p) is known as the . (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
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
11H Orthogonal Projection of a Vector
The orthogonal projection of one vector along another.
From playlist Linear Algebra
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
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
Density of Eigenvalues in a Quasi-Hermitian Random Matrix Model by Joshua Feinberg
Non-Hermitian Physics - PHHQP XVIII DATE: 04 June 2018 to 13 June 2018 VENUE:Ramanujan Lecture Hall, ICTS Bangalore Non-Hermitian Physics-"Pseudo-Hermitian Hamiltonians in Quantum Physics (PHHQP) XVIII" is the 18th meeting in the series that is being held over the years in Quantum Phys
From playlist Non-Hermitian Physics - PHHQP XVIII
Philippe michel - 2/4 Automorphic forms for GL(2)
Philippe michel - Automorphic forms for GL(2)
From playlist École d'été 2014 - Théorie analytique des nombres
Alessandra Sarti: Topics on K3 surfaces - Lecture 3: Basic properties of K3 surfaces
Abstract: Aim of the lecture is to give an introduction to K3 surfaces, that are special algebraic surfaces with an extremely rich geometry. The most easy example of such a surface is the Fermat quartic in complex three-dimensional space. The name K3 was given by André Weil in 1958 in hono
From playlist Algebraic and Complex Geometry
Math 060 Fall 2017 111517C Orthonormal Bases, Orthogonal Matrices, and Method of Least Squares
Definition of orthogonal matrices. Example: rotation matrix. Properties: Q orthogonal if and only if its transpose is its inverse. Q orthogonal implies it is an isometry; that it is isogonal (preserves angles). Theorem: How to find, given a vector in an inner product space, the closest
From playlist Course 4: Linear Algebra (Fall 2017)
Find an Orthogonal Projection of a Vector Onto a Line Given an Orthogonal Basis (R2)
This video explains how t use the orthogonal projection formula given subset with an orthogonal basis. The distance from the vector to the line is also found.
From playlist Orthogonal and Orthonormal Sets of Vectors
Andy Wathen: Parallel preconditioning for time-dependent PDEs and PDE control
We present a novel approach to the solution of time-dependent PDEs via the so-called monolithic or all-at-once formulation. This approach will be explained for simple parabolic problems and its utility in the context of PDE constrained optimization problems will be elucidated. The underlyi
From playlist Numerical Analysis and Scientific Computing
Tathagata Basak: A monstrous(?) complex hyperbolic orbifold
I will report on progress with Daniel Allcock on the ”Monstrous Proposal”, namely the conjecture: Complex hyperbolic 13-space, modulo a particular discrete group, and with orbifold structure changed in a simple way, has fundamental group equal to (MxM)(semidirect)2, where M is the Monster
From playlist Topology
Linear Algebra 7.3 Quadratic Forms
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
Effective bounds for the least solutions of homogeneous quadratic... - Thomas Hille
Special Dynamics Seminar Topic: Effective bounds for the least solutions of homogeneous quadratic Diophantine inequalities Speaker: Thomas Hille Affiliation: Yale University Date: November 15, 2019 For more video please visit http://video.ias.edu
From playlist Mathematics
11J Orthogonal Projection of a Vector
The orthogonal projection of one vector along another.
From playlist Linear Algebra
Quadratic forms and homogeneous dynamics by Anish Ghosh
Probabilistic Methods in Negative Curvature ORGANIZERS: Riddhipratim Basu, Anish Ghosh and Mahan Mj DATE: 11 March 2019 to 22 March 2019 VENUE: Madhava Lecture Hall, ICTS, Bangalore The focal area of the program lies at the juncture of three areas: Probability theory of random wa
From playlist Probabilistic Methods in Negative Curvature - 2019
Orthogonal Set of Functions ( Fourier Series )
Thanks to all of you who support me on Patreon. You da real mvps! $1 per month helps!! :) https://www.patreon.com/patrickjmt !! Orthogonal Set of Functions ( Fourier Series ). Here I give the definition of an orthogonal set of functions and show a set of functions is an orthogonal set.
From playlist All Videos - Part 1
Degenerations of Kahler forms on K3 surfaces, and some dynamics - Simion Filip
Joint IAS/Princeton/Montreal/Paris/Tel-Aviv Symplectic Geometry Topic: Degenerations of Kahler forms on K3 surfaces, and some dynamics Speaker: Simion Filip Date: June 04, 2021 K3 surfaces have a rich geometry and admit interesting holomorphic automorphisms. As examples of Calabi-Yau ma
From playlist Mathematics
11I Orthogonal Projection of a Vector
The Orthogonal Projection of one vector along another.
From playlist Linear Algebra