Matrix decompositions | Lie groups | Matrix theory | Operator theory

Polar decomposition

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).

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Polar Decomposition

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

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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

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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

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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)

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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

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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

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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

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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

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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

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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

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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

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Colleen Robles

https://www.math.ias.edu/files/media/agenda.pdf More videos on http://video.ias.edu

From playlist Mathematics

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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

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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)

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Complex eigenvalues

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

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Singular Value Decomposition

Singular values. The Singular Value Decomposition.

From playlist Linear Algebra Done Right

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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

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