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orthonormal

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Images

  • bangle lumber1 jpg 13 Apr 2007 22 54 51k banana gif 13 Apr 2007 22 54 3k Orthonormal jpg 13 Apr 2007 22 55 52k NY jpg 13 Apr 2007 22 55 70k
  • > > par mfrow = c 1 1 > barplot attr w values > > a phylogenetic orthobasis > data njplot > phy < newick2phylog njplot$tre > wA < phy$Ascores > wW < phy$Wscores > table phylog phylog = phy wA clabel row = 0 clabel col = 0 5
  • orthobasis > data mafragh > w < orthobasis neig mafragh$neig > par mfrow = c 4 2 > for k in 1 8 + s value mafragh$xy w k cleg = 0 sub = as character k + csub = 3 > > par mfrow = c 1 1 > barplot attr w values > > a phylogenetic orthobasis > data njplot > phy < newick2phylog njplot$tre > wA < phy$Ascores > wW < phy$Wscores > table
  • > plot ts ortho 23 start = 1821 end = 1934 freq = 1 ylab = score 23 > par mfrow = c 1 1 > > > >
  • orthonormal basis
  • Obs Std Obs Pvalue 1 R2Max 0 3277626 14 868670 0 001 2 SkR2k 25 5239344 7 200245 1 000 3 Dmax 0 5797304 11 448028 0 001 4 SCE 13 2741429 43 655046 0 001 other elements NULL > attributes lynx $tsp 1 1821 1934 1 > par mfrow = c 2 1 > par mar = c 4 4 2 2 > plot ts lynx > plot ts ortho 23 start = 1821 end = 1934 freq = 1 ylab = score 23
  • Obs Std Obs Pvalue 1 R2Max 0 3566524 0 7936064 0 238 2 SkR2k 5 3204632 2 3237397 0 988 3 Dmax 0 4273104 2 5223580 0 016 4 SCE 1 0474954 2 8965026 0 020 other elements NULL > > a spatial example > data irishdata > neig1 < neig mat01 = 1 irishdata$link > 0 > sco1 < scores neig neig1 > z < scalewt irishdata$tab$cow > orthogram z sco1 class krandtest
  • Obs Std Obs Pvalue 1 R2Max 0 3219990 1 345968 0 095 2 SkR2k 4 0552287 3 979705 1 000 3 Dmax 0 6286936 4 970920 0 001 4 SCE 3 8344874 11 260082 0 001 other elements NULL > > a temporal example > data arrival > w < orthobasis circ 24 > orthogram arrival$hours w class krandtest Monte Carlo tests Call orthogram x = arrival$hours orthobas = w Test
  • > > a phylogenetic orthobasis > data njplot > phy < newick2phylog njplot$tre > wA < phy$Ascores > wW < phy$Wscores > table phylog phylog = phy wA clabel row = 0 clabel col = 0 5 > table phylog phylog = phy wW clabel row = 0 clabel col = 0 5 > > > > >
  • Obs Std Obs Pvalue 1 R2Max 0 7375324 7 434652 0 001 2 SkR2k 4 4263014 4 045642 1 000 3 Dmax 0 6505764 5 554063 0 001 4 SCE 3 5301360 11 870262 0 001 other elements NULL > par mfrow = c 1 2 > dotcircle arrival$hours > dotcircle w 2
  • mm26 equalization gif 27 Jun 1999 02 04 99K mm27 basis images JPG 27 Jun 1999 02 23 29K mm27a orthonormal trans jpg 03 Jan 2000 21 57 142K mm27b complete basis jpg 03 Jan 2000 22 02 147K
  • >> plot x real y 1 length x r >> plot imag y 1 length x y which produces the figure below
  • newick2phylog ungulates$tre > FemBodyMass < log ungulates$tab 1 > NeonatBodyMass < log ungulates$tab 2 +ungulates$tab 3 2 > plot FemBodyMass NeonatBodyMass pch = 20 cex = 2 > abline lm NeonatBodyMass~FemBodyMass > z < residuals lm NeonatBodyMass~FemBodyMass > dotchart phylog ung phy val = z clabel n = 1 + labels n = ung phy$Blabels cle = 1 5 cdot
  • k in 1 31 + plot w k type= S xlab = ylab = xaxt = n + yaxt = n xaxs = i yaxs = i ylim=c 4 5 4 5 + points w k type = p pch = 20 cex = 1 5 + > > a 1D orthobasis > w < orthobasis line n = 33 > par mfrow = c 8 4 > par mar = c 0 1 0 1 0 1 0 1 > for k in 1 32 + plot w k type= l xlab = ylab = xaxt = n
  • > table phylog phylog = phy wW clabel row = 0 clabel col = 0 5 > > > > >
  • > abline lm NeonatBodyMass~FemBodyMass > z < residuals lm NeonatBodyMass~FemBodyMass > dotchart phylog ung phy val = z clabel n = 1 + labels n = ung phy$Blabels cle = 1 5 cdot = 2 > table phylog ung phy$Bscores ung
  • > dotcircle w 2 > par mfrow = c 1 1 > > data lynx > ortho < orthobasis line 114 > orthogram lynx ortho class krandtest Monte Carlo tests Call orthogram x = lynx orthobas = ortho Test number 4
  • > par mfrow = c 1 1 > barplot attr w$orthobasis values > > Haar 1D orthobasis > w < orthobasis haar 32 > par mfrow = c 8 4 > par mar = c 0 1 0 1 0 1 0 1 > for k in 1 31 + plot w k type= S xlab = ylab = xaxt = n
  • kstudy2608263 0001 jpg
  • > attributes lynx $tsp 1 1821 1934 1 > par mfrow = c 2 1 > par mar = c 4 4 2 2 > plot ts lynx > plot ts ortho 23 start = 1821 end = 1934 freq = 1 ylab = score 23 > par mfrow = c 1 1 > > > >
  • > > par mfrow = c 1 1 > barplot attr w values > > w < orthobasis circ n = 26 > par mfrow = c 5 5 > par mar = c 0 1 0 1 0 1 0 1 > for k in 1 25 > dotcircle w k xlim = c 1 5 1 5 cleg = 0 > > par mfrow = c 1 1
  • ans = 0 3022 Indeed the frame operator is very close to identity as seen below coef is a length L cell array where L is the number of stages Each element of the cell array holds another q by q cell array >> coef
  • Obs Std Obs Pvalue 1 R2Max 0 3566524 0 7176604 0 254 2 SkR2k 5 3204632 2 2915694 0 990 3 Dmax 0 4273104 2 5559216 0 013 4 SCE 1 0474954 2 6835207 0 026 other elements NULL > orthogram z phyl=ung phy the same thing class krandtest Monte Carlo tests Call orthogram x = z phylog = ung phy Test number 4 Permutation number 999 Alternative hypothesis
  • moi je dis si AB est perpendiculaire a AD et AB=AD on dit alors que le repére A vecteur AB vecteur AD est un repere orthonormal merci de m aider es ke cest bon ce que jai mis merci
  • k in 1 32 + plot w k type= l xlab = ylab = xaxt = n + yaxt = n xaxs = i yaxs = i ylim=c 1 5 1 5 + points w k type = p pch = 20 cex = 1 5 + > > par mfrow = c 1 1 > barplot attr w values > > w < orthobasis circ n = 26 > par mfrow = c 5 5 > par mar = c 0 1 0 1 0 1 0 1 > for k in 1 25 > dotcircle w k
  • Repère orthonormal
  • = c 4 4 > w < gridrowcol 8 8 > for k in 1 16 + s value w$xy w$orthobasis k cleg = 0 csi = 2 incl = FALSE + addax = FALSE sub = k csub = 4 ylim = c 0 10 cgri = 0 > par mfrow = c 1 1 > barplot attr w$orthobasis values > > Haar 1D orthobasis > w < orthobasis haar 32 > par mfrow = c 8 4 > par mar = c 0 1 0 1 0 1 0 1 > for k in 1 31
  • error plotted along with the input signal via >> plot x hold on plot x y 1 length x r producing the figure below The signal is plotted in blue and reconstruction error in red Now let s turn to overcomplete DFT Modulated filter banks Given a p q pair this is an FB as shown below where W = exp j2pi q
  • orthnorm jpg
  • >> coefz 3 2 1 round end 2 round end 2 = 1 >> di = IDIRDWT2D coefz h p q Repeating this see demo m we get the following synthesis functions
  • but in a fancier way To be consistent with Jozef Cohen s discoveries and enjoy other benefits we can make color vectors by starting with Orthonormal Opponent Color Matching Functions The functions are easy to generate or can be found at http www jimworthey com orthobasis txt
  • kstudy2608263 0002 jpg
  • 1 004585982218658 >> The resulting filter is shown below
  • > table phylog ung phy$Bscores ung phy clabel n = 1 + labels n = ung phy$Blabels > orthogram z ung phy$Bscores class krandtest Monte Carlo tests Call orthogram x = z orthobas = ung phy$Bscores Test number 4 Permutation number 999 Alternative hypothesis
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  • 線性代數第四個基本定理即奇異值分解 其核心價值在於它為矩陣變換
  • orthobasis png
  • > par mfrow = c 1 2 > dotcircle arrival$hours > dotcircle w 2 > par mfrow = c 1 1 > > data lynx > ortho < orthobasis line 114 > orthogram lynx ortho class krandtest Monte Carlo tests Call orthogram x = lynx orthobas =
  • > z < residuals lm NeonatBodyMass~FemBodyMass > dotchart phylog ung phy val = z clabel n = 1 + labels n = ung phy$Blabels cle = 1 5 cdot = 2 > table phylog ung phy$Bscores ung phy clabel n = 1 + labels n = ung phy$Blabels > orthogram z ung phy$Bscores class krandtest Monte Carlo tests Call orthogram x = z orthobas
  • Vector Expansion using Base Vectors 1 png 04 Sep 2003 16 39 41K Vector Expansion using Base Vectors pdf 23 Aug 2005 13 55 128K Vector Algebra using Orthonormal Base Vectors 1 png 04 Sep 2003 16 42 38K Vector Algebra using Orthonormal Base Vectors pdf 23 Aug 2005 14 31 594K

Videos

  • Example of Diagonalizing a Symmetric Matrix (Spectral Theorem) Linear Algebra: For the real symmetric matrix [3 2 / 2 3], 1) verify that all eigenvalues are real, 2) show that eigenvectors for distinct eigenvalues are orthogonal with respect to the standard inner product, and 3) find an orthogonal matrix P such that P^{-1}AP = D is diagonal. The Spectral Theorem states that every symmetric matrix can be put into real diagonal form using an orthogonal change of basis matrix (or there is an orthonormal basis of eigenvectors).
  • Gram-Schmidt with matrices We orthogonalize a set of matrices, then normalize them, then find the Fourier coefficients for a given matrix.
  • Math 2800 Checkpoint Quiz 20 Part 1 We apply a Gram Schmidt process to a given basis in a given inner product space.
  • Lec 1 | MIT 6.451 Principles of Digital Communication II Introduction Sampling Theorem and Orthonormal PAM/QAM Capacity of AWGN Channels View the complete course: ocw.mit.edu License: Creative Commons BY-NC-SA More information at ocw.mit.edu More courses at ocw.mit.edu
  • 2554 Math 3 lecture 6 Ch 6-3 orthonormal bases and Gram-Schmidt process.avi Inner Product spaces, Orthonormal bases, Gram-Schmidt process, normalization, orthogonal projections.
  • Representation Theory - Part 9 - Schur Orthogonality Relations Representation Theory of Finite Groups: As a first step to Fourier ***ysis on finite groups, we state and prove the Schur Orthogonality Relations. With these relations, we may form an orthonormal basis of matrix coefficients for L^(G), the set of functions on G. We also define characters and state the corresponding SORs.
  • Linear Algebra: Coordinates with respect to orthonormal bases (Bangla)
  • frame-part2.wmv Introduction to frame theory. Starts with Hilbert space definition and concludes with intro to Frame operators.
  • Linear Algebra: Introduction to Orthonormal Bases (Bangla)
  • Linear Algebra: Coordinates with respect to orthonormal bases Seeing that orthonormal bases make for good coordinate systems
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  • Linear Algebra Introduction to Orthonormal Bases
  • Topics In Quantum Mechanics Video #7: Example Of Orthonormal Function Hundreds of Free Problem Solving Videos And FREE REPORTS from www.digital-
  • Example of Gram-Schmidt Orthogonalization Linear Algebra: Construct an orthonormal basis of R^3 by applying the Gram-Schmidt orthogonalization process to (1, 1, 1), (1, 0, 1), and (1, 1, 0). In addition, we show how the Gram-Schmidt equations allow one to factor an invertible matrix into an orthogonal matrix times an upper triangular matrix.
  • Soaring in Hilbert Space.mpg Eagles are soaring in 3-dimensional space but mathematicians were soaring beyond 3-dimensional space to end up with infinite dimensionality. Since the area of greek mathematics (the time of the famous mathematician Pythagoras, about 2500 years ago) infinity has been figured out to be a wild animal which does not allow to be tamed. Thus it took about 2500 years until Georg Cantor came up with his foundation of set theory, which gave modern mathematics a complete new basis and allowed finally the taming of infinity. Based on Georg Cantor's work another famous mathematician, David Hilbert, developed his powerful theory of function space, where one of his most powerful creations is now known as Hilbert space. No wonder that Hilbert space is the mathematical basis of quantum mechanics since David Hilbert was permanently in direct touch with quantum physicsists during the evolution of quantum mechanics. In a metaphorical language we could conclude that eagles might feel infinity by soaring in their three dimensional space but Hilbert space is a tremendously richer environment which allows mathematicians and physicsists to soar in infinite dimensional space.
  • Fourier Series: Example of Orthonormal Set of Functions Differential Equations: Prelude to Fourier series. Show that the sets B1 = {1, sqrt(2) cos(x), sqrt(2) sin(x)} and B2 = {1, exp(-ix), exp(ix)} are orthonormal sets of functions with respect to the inner product (f, g) = 1/2pi int f(x) {bar g(x)} dx. Then verify Parseval's Identity for f(x) = sin(x) with respect to each set.
  • Lin Alg: Finding projection onto subspace with orthonormal basis example Example of finding the transformation matrix for the projection onto a subspace with an orthonormal basis
  • Linear Algebra - Orthonormal Bases Watch more at Other subjects include Calculus, Statistics, Biology, Chemistry, Physics, Organic Chemistry, Computer Science, Algebra 1/2, Basic Math, Pre Calculus, Geometry, and Pre Algebra. -All lectures are broken down by individual topics -No more wasted time -Just search and jump directly to the answer
  • Lin Alg: Finding projection onto subspace with orthonormal basis example (Bangla)
  • Oct27-c-OrthonormalBasis1 Lecture on Oct. 27, 2009. Orthonormal basis (part 1 of 3).
  • Distinguishing quantum states Explains the result that it is always possible to distinguish orthonormal quantum states, and how this lets us better understand the superdense coding protocol. Part of a series on "Quantum computing for the determined". The full series is at:
  • Lecture 4 | Introduction to Linear Dynamical Systems Professor Stephen Boyd, of the Electrical Engineering department at Stanford University, lectures on orthonormal sets of vectors and QR factorization for the course, Introduction to Linear Dynamical Systems (EE263). Introduction to applied linear algebra and linear dynamical systems, with applications to circuits, signal processing, communications, and control systems. Topics include: Least-squares aproximations of over-determined equations and least-norm solutions of underdetermined equations. Symmetric matrices, matrix norm and singular value decomposition. Eigenvalues, left and right eigenvectors, and dynamical interpretation. Matrix exponential, stability, and asymptotic behavior. Multi-input multi-output systems, impulse and step matrices; convolution and transfer matrix descriptions. Complete Playlist for the Course: EE 263 Course Website: www.stanford.edu Stanford University: www.stanford.edu Stanford University Channel on YouTube:
  • Orthonormality of Standing Waves TheWolfram Demonstrations Project contains thousands of free interactive visualizations, with new entries added daily. This Demonstration shows two standing waves and their inner product, checking the orthonormality condition SubsuperscriptBox[?, 0, L]f_m(x) f_n(x)d?x=?_m,n. Each normalized wave has the form f_k(x)=SqrtBox[FractionBox[2, L]]sin( ( k ? x ) / ( L ) ) (wit... Contributed by: Porscha McRobbie and Eitan Geva
  • 2554 Math 3 lecture 7 Ch 6 review inner product space and orthogonal basis.avi Linear Algebra, inner product space, review, orthogonal, orthonormal basis.
  • Oct27-d-OrthonormalBasis2 Lecture on Oct. 27, 2009. Orthonormal basis (part 2 of 3).
  • Lin Alg Finding projection onto subspace with orthonormal basis example
  • Orthogonal, orthonormal, coordinates for orthogonal basis As title says.
  • Example of Spectral Theorem (3x3 Symmetric Matrix) Linear Algebra: We verify the Spectral Theorem for the 3x3 real symmetric matrix A = [ 0 1 1 / 1 0 1 / 1 1 0 ]. That is, we show that the eigenvalues of A are real and that there exists an orthonormal basis of eigenvectors. In other words, we can put A in real diagonal form using an orthogonal matrix P. (Eigenvalues and eigenvectors for this A are found in the video "Eigenvalues and Eigenvectors.")
  • Fourier Series: Example of Parseval's Identity Differential Equations: Find the Fourier coefficients of the square wave function f(x) = -1 on the interval (-pi, 0), 1 on the interval (0, pi). Then state Parseval's Identity in this case. With this, we show that sum 1/n^2 = pi^2/6.
  • Gram-Schmidt Process in Two Dimensions The Wolfram Demonstrations Project contains thousands of free interactive visualizations, with new entries added daily. The Gram-Schmidt process is a means for converting a set of linearly independent vectors into a set of orthonormal vectors. If the set of vectors spans the ambient vector space, then this produces an orthonormal basis for the vector space. The Gram-Schm... Contributed by: Chris Boucher
  • frame-theory-part1.wmv Introduction to frame theory. Starts with Hilbert space definition and concludes with intro to Frame operators.
  • Video Tutorial: Finding Orthogonal and Orthonormal Bases This short tutorial demonstrates how to form an orthogonal or orthonormal basis in Maple, given a set of vectors. For more information, visit us at:
  • Example of Fourier's Trick Linear Algebra: Given an orthonormal basis of R^n, we present a quick method for finding coefficients of linear combination in terms of the basis. We also give an ***ogue of Parseval's Identity, which relates these coefficients to the squared length of the vector.
  • Lin Alg Projections onto subspaces with orthonormal bases
  • Econometrics I Lecture 03d Vector Space, OrthoNormal Vectors, Norm Lecture by Dr. Andrew Buck, Professor of Economics, Temple University, Philadelphia, PA, USA.
  • One Advantage of Using an Orthonormal Basis
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  • Lec 10 | MIT 6.450 Principles of Digital Communications I, Fall 2006 Lecture 10: Degrees of freedom, orthonormal expansions, and aliasing View the complete course at: ocw.mit.edu License: Creative Commons BY-NC-SA More information at ocw.mit.edu More courses at ocw.mit.edu
  • Partial measurements in an arbitrary basis Explains what happens when you measure just part of a quantum system, in an arbitrary orthonormal basis. This is the final concept needed for our discussion of quantum teleportation. Part of a series on "Quantum computing for the determined". The full series is at:
  • Spectral Theorem for Real Matrices: General 2x2 Case Linear Algebra: We state and prove the Spectral Theorem for a real 2x2 symmetric matrix A = [ab \ bc]. That is, we show that the eigenvalues of A are real and that there exists an orthonormal basis of eigenvectors for A.
  • Top Ten Theorems: Metric Geometry Part 3 Theorems #4, #3, #2, #1 count down completed. Contraction of a rank two tensor to an invariant scalar. Derivation of the General Relativistic field equations for empty space. The arc length integral in n-dimensions. Properties of a diagonal metric tensor. Covariant and contravariant unit vectors, and the ortho-normal basis set.
  • Linear Algebra: Introduction to Orthonormal Bases Looking at sets and bases that are orthonormal -- or where all the vectors have length 1 and are orthogonal to each other.