Matrix decompositions | Lie groups | Matrix theory | Operator theory
In mathematics, the polar decomposition of a square real or complex matrix is a factorization of the form , where is a unitary matrix and is a positive semi-definite Hermitian matrix, both square and of the same size. Intuitively, if a real matrix is interpreted as a linear transformation of -dimensional space , the polar decomposition separates it into a rotation or reflection of , and a scaling of the space along a set of orthogonal axes. The polar decomposition of a square matrix always exists. If is invertible, the decomposition is unique, and the factor will be positive-definite. In that case, can be written uniquely in the form , where is unitary and is the unique self-adjoint logarithm of the matrix . This decomposition is useful in computing the fundamental group of (matrix) Lie groups. The polar decomposition can also be defined as where is a symmetric positive-definite matrix with the same eigenvalues as but different eigenvectors. The polar decomposition of a matrix can be seen as the matrix analog of the polar form of a complex number as , where is its absolute value (a non-negative real number), and is a complex number with unit norm (an element of the circle group). The definition may be extended to rectangular matrices by requiring to be a semi-unitary matrix and to be a positive-semidefinite Hermitian matrix. The decomposition always exists and is always unique. The matrix is unique if and only if has full rank. (Wikipedia).
The analogy between the complex numbers and L(V). The Polar Decomposition: If T is an operator on a finite-dimensional inner product space V, then there exists an isometry on V such that T equals S times the square root of T*T.
From playlist Linear Algebra Done Right
Seth Lloyd - Quantum polar decomposition - IPAM at UCLA
Recorded 25 January 2022. Seth Lloyd of the Massachusetts Institute of Technology presents "Quantum polar decomposition" at IPAM's Quantum Numerical Linear Algebra Workshop. Abstract: The polar decomposition decomposes a matrix into the product of a unitary and an Hermitian matrix. This ta
From playlist Quantum Numerical Linear Algebra - Jan. 24 - 27, 2022
Polar to rectangular equation conversion
Learn how to convert between rectangular and polar equations. A rectangular equation is an equation having the variables x and y which can be graphed in the rectangular cartesian plane. A polar equation is an equation defining an algebraic curve specified by r as a function of theta on the
From playlist Convert Between Polar/Rectangular (Equations) #Polar
Polar Coordinates and Graphing Polar Equations
Everything we have done on the coordinate plane so far has been using rectangular coordinates. That's the x and y we are used to. But that's not the only coordinate system. We can also use polar coordinates, which graph points in terms of a radius, or distance from a pole, and theta, the a
From playlist Mathematics (All Of It)
Linear Algebra 23a: Polar Decomposition - A Product of an Orthogonal and Symmetric Matrices
https://bit.ly/PavelPatreon https://lem.ma/LA - Linear Algebra on Lemma http://bit.ly/ITCYTNew - Dr. Grinfeld's Tensor Calculus textbook https://lem.ma/prep - Complete SAT Math Prep
From playlist Part 3 Linear Algebra: Linear Transformations
Converting the Rectangular Equation x^2 + y^2 = 4 into Polar Form
Please Subscribe here, thank you!!! https://goo.gl/JQ8Nys Converting the Rectangular Equation x^2 + y^2 = 4 into Polar Form
From playlist Trigonometry
Converting a linear equation to polar form
Learn how to convert between rectangular and polar equations. A rectangular equation is an equation having the variables x and y which can be graphed in the rectangular cartesian plane. A polar equation is an equation defining an algebraic curve specified by r as a function of theta on the
From playlist Convert Between Polar/Rectangular (Equations) #Polar
Rectangular to polar equation conversion
Learn how to convert between rectangular and polar equations. A rectangular equation is an equation having the variables x and y which can be graphed in the rectangular cartesian plane. A polar equation is an equation defining an algebraic curve specified by r as a function of theta on the
From playlist Convert Between Polar/Rectangular (Equations) #Polar
Mumford-Tate Groups and Domains - Phillip Griffiths
Phillip Griffiths Professor Emeritus, School of Mathematics March 28, 2011 For more videos, visit http://video.ias.edu
From playlist Mathematics
Inaugural Imaging & Inverse Problems (IMAGINE) OneWorld SIAM-IS Virtual Seminar Series Talk
Date: Wednesday, October 14, 10:00am EDT Speaker: Michael Friedlander, University of British Columbia Title: Polar deconvolution of mixed signals Abstract: The signal demixing problem seeks to separate the superposition of multiple signals into its constituent components. We model the s
From playlist Imaging & Inverse Problems (IMAGINE) OneWorld SIAM-IS Virtual Seminar Series
Write a rectangular equation in polar form
Learn how to convert between rectangular and polar equations. A rectangular equation is an equation having the variables x and y which can be graphed in the rectangular cartesian plane. A polar equation is an equation defining an algebraic curve specified by r as a function of theta on the
From playlist Convert Between Polar/Rectangular (Equations) #Polar
https://www.math.ias.edu/files/media/agenda.pdf More videos on http://video.ias.edu
From playlist Mathematics
Determinantal varieties and asymptotic expansion of Bergman kernels by Harald Upmeier
DISCUSSION MEETING ANALYTIC AND ALGEBRAIC GEOMETRY DATE:19 March 2018 to 24 March 2018 VENUE:Madhava Lecture Hall, ICTS, Bangalore. Complex analytic geometry is a very broad area of mathematics straddling differential geometry, algebraic geometry and analysis. Much of the interactions be
From playlist Analytic and Algebraic Geometry-2018
Relativistic Spin Hydrodynamics by Amaresh Jaiswal
DISCUSSION MEETING EXTREME NONEQUILIBRIUM QCD (ONLINE) ORGANIZERS: Ayan Mukhopadhyay (IIT Madras) and Sayantan Sharma (IMSc Chennai) DATE & TIME: 05 October 2020 to 09 October 2020 VENUE: Online Understanding quantum gauge theories is one of the remarkable challenges of the millennium
From playlist Extreme Nonequilibrium QCD (Online)
Matrix with complex eigenvalues and diagonalization. Featuring polar decomposition, which is like polar coordinates, but for matrices. Check out my Eigenvalues playlist: https://www.youtube.com/watch?v=H-NxPABQlxI&list=PLJb1qAQIrmmC72x-amTHgG-H_5S19jOSf Subscribe to my channel: https://w
From playlist Eigenvalues
Singular values. The Singular Value Decomposition.
From playlist Linear Algebra Done Right
Solve a System of Linear Equations Using LU Decomposition
This video explains how to use LU Decomposition to solve a system of linear equations. Site: http://mathispower4u.com Blog: http://mathispower4u.wordpress.com
From playlist Matrix Equations