Integral transforms | Signal processing | Time–frequency analysis | Hamiltonian mechanics | Fourier analysis
In Hamiltonian mechanics, the linear canonical transformation (LCT) is a family of integral transforms that generalizes many classical transforms. It has 4 parameters and 1 constraint, so it is a 3-dimensional family, and can be visualized as the action of the special linear group SL2(R) on the time–frequency plane (domain). As this defines the original function up to a sign, this translates into an action of its double cover on the original function space. The LCT generalizes the Fourier, fractional Fourier, Laplace, Gauss–Weierstrass, Bargmann and the Fresnel transforms as particular cases. The name "linear canonical transformation" is from canonical transformation, a map that preserves the symplectic structure, as SL2(R) can also be interpreted as the symplectic group Sp2, and thus LCTs are the linear maps of the time–frequency domain which preserve the symplectic form, and their action on the Hilbert space is given by the Metaplectic group. The basic properties of the transformations mentioned above, such as scaling, shift, coordinate multiplication are considered. Any linear canonical transformation is related to affine transformations in phase space, defined by time-frequency or position-momentum coordinates. (Wikipedia).
Showing something is a linear transformation Check out my Linear Equations playlist: https://www.youtube.com/playlist?list=PLJb1qAQIrmmD_u31hoZ1D335sSKMvVQ90 Subscribe to my channel: https://www.youtube.com/channel/UCoOjTxz-u5zU0W38zMkQIFw
From playlist Linear Transformations
Linear Transformations and Linear Systems
In this video we discuss linear transformations. We start by examining the mathematical definition of a linear transformation and apply it to several examples including matrix multiplication and differentiation. We then see how linear transformations relate to linear systems (AKA linear
From playlist Linear Algebra
Math 060 101817C Matrix Transformations of Linear Transformations
Recall: If linear transformations agree on a basis, then they are equal. Recall: To each matrix A corresponds a linear transformation (left-multiplication by A). Theorem: Every linear transformation between Euclidean spaces corresponds to left-multiplication by some matrix. Example. Ex
From playlist Course 4: Linear Algebra (Fall 2017)
Linear Transformations: One-One
Linear Algebra: We recall the definition of one-one for functions and apply it to linear transformations. We obtain a simple rule for checking one-one in this case: either the kernel is zero or the associated matrix has a pivot in each column in row echelon form. Several examples are gi
From playlist MathDoctorBob: Linear Algebra I: From Linear Equations to Eigenspaces | CosmoLearning.org Mathematics
2.2.2 What is a linear transformation?
2.2.2 What is a linear transformation?
From playlist LAFF - Week 2
Sketch a Linear Transformation of a Unit Square Given the Transformation Matrix (Shear)
This video explains 2 ways to graph a linear transformation of a unit square on the coordinate plane.
From playlist Matrix (Linear) Transformations
Math 060 102317 Matrix Representations of Linear Transformations III
Review of the construction of the matrix representation of a linear transformation with respect to bases. Review: matrix representation of a linear transformation between Euclidean spaces with respect to bases (not necessarily the standard bases). Example.
From playlist Course 4: Linear Algebra (Fall 2017)
Walter CRAIG - Birkhoff normal form for nonlinear wave equations
Many theorems on global existence of small amplitude solutions of nonlinear wave equations in ${\mathbb R}^n$ depend upon a competition between the time decay of solutions and the degree of the nonlinearity. Decay estimates are more effective when inessential nonlinea
From playlist Trimestre "Ondes Non Linéaires" - May Conference
Lecture 9 | Modern Physics: Classical Mechanics (Stanford)
Lecture 9 of Leonard Susskind's Modern Physics course concentrating on Classical Mechanics. Recorded December 20, 2007 at Stanford University. This Stanford Continuing Studies course is the first of a six-quarter sequence of classes exploring the essential theoretical foundations of mo
From playlist Course | Modern Physics: Classical Mechanics
Let's Learn Physics: A Surprise to Be Sure, but a Welcome One
In this stream, we will look at how to incorporate more general "canonical transformations" into our Hamiltonian framework. Along the way, we will see some very deep relations between transformations, symmetries, and conservation, as well as some other interesting relations between paramet
From playlist Let's Learn (Classical) Physics: ZAP Physics Livestreams
Lecture 6: Gauge-equivariant Mesh CNN - Pim de Haan
Video recording of the First Italian School on Geometric Deep Learning held in Pescara in July 2022. Slide: https://www.sci.unich.it/geodeep2022/slides/2022-07-27%20Mesh%20-%20First%20Italian%20GDL%20School.pdf
From playlist First Italian School on Geometric Deep Learning - Pescara 2022
Change of basis - transformation matrices
Please, subscribe or support on Steady: https://steadyhq.com/en/brightsideofmaths Then you can see when I'm doing a live stream. Here, I present some visualisation and calculation for the change of basis and transformation matrices. I apologise for my pronunciation. The focus is on the ma
From playlist Linear algebra (English)
Matrix Transformations are the same thing as Linear Transformations
Learning Objectives: 1) Recall the defining properties of Matrix-vector product and of Linear Transformations 2) Apply algebraic rules to deduce that Matrix transformations are Linear transformations 3) Prove that Linear Transformations are Matrix transformations by writing a vector as a l
From playlist Linear Algebra (Full Course)
Soft Factorization in Non-Abelian Gauge Theories by Prahar Mitra
PROGRAM RECENT DEVELOPMENTS IN S-MATRIX THEORY (ONLINE) ORGANIZERS: Alok Laddha, Song He and Yu-tin Huang DATE: 20 July 2020 to 31 July 2020 VENUE:Online Due to the ongoing COVID-19 pandemic, the original program has been canceled. However, the meeting will be conducted through online
From playlist Recent Developments in S-matrix Theory (Online)