Orthogonality Definition and 167 Threads

Let V be a vector space over a field h and let n be a positive integer. Let f:V -> h^n be a linear map given by f(v) = (f1(v), f2(v), ..., fn(v)). Call two vectors (g1, ..., gn) and (h1, ..., hn) in h^n "orthogonal" if g1 h1 + ... + gn hn = 0 Suppose the only vector orthogonal to every...

Homework Statement Denote the inner product of f,g \in H by <f,g> \in R where H is some(real-valued) vector space a) Explain linearity of the inner product with respect to f,g. Define orthogonality. b) Let f(x) and g(x) be 2 real-valued vector functions on [0,1]. Could the inner product be...

Say I have 2 complex (normalized) column vectors x and y in C^N: The standard dot product <x,y> = x*y (where * denotes conjugate transpose) gives me a "measure of orthogonality" of the two vectors. Now the bilinear product (c,y) = x'y (' denotes transpose) seems to give another "measure of...

Could someone kindly explain whether the 90 degree phase difference between sine & cosine functions contribute to the fact that they are orthogonal? I just studied Fourier series and treating sines and cosines as vectors is fine for my brain to handle, but I can't tell whether the phase...

Homework Statement There is a recursion relation between the Legendre polynomial. To see this, show that the polynomial x p_k is orthogonal to all the polynomials of degree less than or equal k-2. Homework Equations <p,q>=0 if and only if p and q are orthogonal. The Attempt at a...

can't figure out how that underlined segment transforms into what's at the bottom: [PLAIN]http://img7.imageshack.us/img7/9493/imag0254p.jpg

According to the orthogonality property of the associated Legendre function P_l^{|m|}(cos\theta) we have that: \int_{0}^{\pi}P_{l}^{|m|}(cos\theta){\cdot}P_{l'}^{|m'|}(cos\theta)sin{\theta}d\theta=\frac{2(l+m)!}{(2l+1)(l-m)!}{\delta}_{ll'} What I am looking for is an orthogonality...

Homework Statement For reference: Problem 1.8.5 parts (3) , R. Shankar, Principles of Quantum Mechanics. Given array \Omega , compute the eigenvalues ( e^i^\theta and e^-^i^\theta ). Then (3) compute the eigenvectors and show that they are orthogonal. Homework Equations Eulers...

a question on orthogonality relating to Fourier analysis and also solutions of PDEs by separation of variables. I've used the fact that the following expression (I chose sine, also cosine works): \int_{0}^{2\pi}\sin mx\sin nxdx equals 0 unless m=n in which case it equals pi in...

Homework Statement A Givens rotation is a matrix J(i,k) that is the identity matrix except jii = jkk = c and jik = -jki = s where c2 + s2 = 1. Let x = [1,-1,3]T. Find the rotation matrix J(2,3) such that the third element of Jx is zero. Show that J(2,3) is orthogonal. Homework Equations To...

I think I finally understand the wedge product & think it explains things in 2-forms that have been puzzling me for a long time. My post consists of the way I see things regarding the wedge product & interspersed with my thoughts are only 3 questions (in bold!) that I'm hoping for some...

Hello there! I have a group represented by the following matricies: \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right)\] ; 0.5\left( \begin{array}{cc} -1 & \sqrt{3} \\ -\sqrt{3} & -1 \end{array} \right)\] and\quad 0.5 \left( \begin{array}{cc} -1 & -\sqrt{3} \\ \sqrt{3} & -1...

Orthogonality and orthonormality ?? Hi What does mean orthogonality and orthonormality physically ? e.g. orthogonal or orthonormal wavefunctions king regards Nawzad A.

Anyone who knows the limits of orthogonality for Bessel polynomials? Been searching the Internet for a while now and I can't find a single source which explicitly states these limits (wiki, wolfram, articles, etc). One thought: since the Bessel polynomials can be expressed as a generalized...

I have two wavefunctions that I need to normalize but I cannot figure out how to get them into an acceptable integrable form... the first is psi=(2-(r/asub0))*e^(-r/asub0) the second is psi=rsin(theta)*cos(phi)*e^(-r/2asub0) I know these need to be in the form (where psi will be name y for...

I want to to know what's the usage of this theory in our life or is there any important application depend on that theory I study it in physics of engineering but i want to to know what's the useful for it?

Homework Statement Show that the orthogonality relation for the "cosine basis functions" used in the Fourier series is 1/L\intcos[(n*pi*x)/L)]cos[(m*pi*x)/L)]dx = {Sin([n-m]*pi)}/[(n-m)*pi] + {Sin([n+m]*pi)}/[(n+m)*pi] By considering the different integer n and m, show that the right...

Homework Statement For spherical coordinates, we will need to use Legendre Polynomials, a.Sketch graphs of the first 3 – P0(x), P1(x), and P2(x). b.Evaluate the orthogonality relationship (eq 3.68) to show these 3 functions are orthogonal to each other. (3 integrals). c.Show that the...

Homework Statement I need to prove the equation attached. I also have to describe why the integrals vanish. Homework Equations The Attempt at a Solution I am not sure how to begin. Our teacher told us this equation is known as the orthogonality condition for sines. I also know...

I'm having trouble understanding why a derivative of a time dependent vector function is orthogonal to the original function. Can anybody give me some enlightenment? I searched around for some previous talk about this, and I can't find anything. Thanks.

Homework Statement Consider L2, the inner product space of the complex sequences x = (xn) such that \sum xi converges, with the inner product given by <x,y> = (sum of) xi yi(complex conjugate) Now let x = (1,0,1,0,1,0,0,0...) y = (1,i,0,i,0,i,0,0,0...) z =...

Hello, I'm trying to show that Integral[x*J0(a*x)*J0(a*x), from 0 to 1] = 1/2 * J1(a)^2 Here, (both) a's are the same and they are a root of J0(x). I.e., J0(a) = 0. I have found and can do the case where you have two different roots, a and b, and the integral evaluates to zero...

Ok so I've been working on this problem and I'm really having some struggles grasping it. Here it is: Let W be some subspace of Rn, let WW consist of those vectors in Rn that are orthognoal to all vectors in W. 1) Show that WW is a subspace of Rn? So for this part I'm thinking that...

Homework Statement If there exists no function, f(x), except zero, with the property that \int_{a}^{b}{\phi_{n}(x)}f(x)w(x)dx=0 for all \phi_{n}, then the set {\phi_{n}(x)} is said to be complete. Write a similar statement expressing the completeness of a set of basis vectors in...

Hello, Let's consider the L^2(\mathbb{R}) space with an inner product, and the complex sinusoids in the interval (-\infty,+\infty). Is it correct to say that the complex sinusoids form an orthogonal basis for this space? One would need to have: \int_{-\infty}^{+\infty}e^{ipx}e^{-iqx}dx=0 for...

Hello, I'd like to prove the orthogonality of two "shifted" Sinc functions, but I can't find the mistake. Here is my attempt: \int_{-\infty}^{+\infty}sinc(x)sinc(x-x_0)dx Observing this quantity can be obtained by evaluating the Fourier transform at zero, we have: \mathcal{F}\{...

Homework Statement If you want to show two wavefunctions are orthogonal, do you have to normalize the wavefunctions first then take the integral of the product and see if they're equal to 0? Homework Equations n/a The Attempt at a Solution not really applicable. I just want a...

Hello I have been wondering for some time about, why I have to use orthogonality properties in a special kind of PDE problem I have encountered a few times now. As an example see exercise 13-3 in this file: http://www.student.dtu.dk/~s072258/01246-2009-week13.pdf" I have described my...

I want to prove orthogonality of associated Legendre polynomial. In my textbook or many posts, \int^{1}_{-1} P^{m}_{l}(x)P^{m}_{l'}(x)dx = 0 (l \neq l') is already proved. But, for upper index m, \int^{1}_{-1} P^{m}_{n}(x)P^{k}_{n}(x)\frac{dx}{ ( 1-x^{2} ) } = 0 (m \neq k) is not...

Hello, As we know, the wave function of infinite potential wells form a complete orthogonal base. I have tried now to solve out the wave function for finite potential well, checking the orthogonality, I found that they are no longer orthogonal to each other (I mean the wave function...

Hi there! In thermal field theory, the Matsubara frequencies are defined by \nu_n = \frac{2n\pi}{\beta} for bosons and \omega_n = \frac{(2n+1)\pi}{\beta} for fermions. Assuming discrete imaginary time with time indices k=0,\hdots,N, it is easy to obtain the following orthogonality relation...

Hi, would anyone be able to explain how to evaluate a function using orthogonality (i.e. using orthogonality to solve a definite integration problem with sines/cosines)? Thank you

Homework Statement a) Let v be a unit vector in R^3 and u be a vector which is orthogonal to v. Show v x (v x u) = -u b) Let v and u be orthogonal unit vectors in R^3. Show u x (v x (v x (v x u))) = -v Homework Equations The Attempt at a Solution I am very lost in this...

Homework Statement I have that the general solution of a function is f(\rho,t)=\Sigmac(m)Jo(\alpha\rho\a) exp[-Dtm^2] where c(m) are constants. I need to find an expression for c(m) in terms of an integral Homework Equations Orthogonality relation given is \intdx x...

How can I show that the binormal vector is orthogonal to the tangent and normal vector. I know i should use the dot product to determine this, however i do i actually go about doing it?

Orthogonality Property of Hyperbolic functions ? Hi all, I have seen Orthogonal property for trigonomeric functions but I am unsure if there is something similar for sinh() , cosh() ? . I know that the integral of inner product of the two functions should be zero for them to be...

Vector Planes & Orthogonality -- Help! I must be doing something really stupid, and I'll kick myself when you point it out, but I'm having difficulty with this question: Find the unit normal to the plane x + 2y – 2z = 15. What is the distance of the plane from the origin? OK, so I know I need...

Homework Statement Find the point on the line y = 2x+1 that is closest to the point (5,2) Homework Equations Vector Projection (x^Ty/y^Ty)*y x and y are orthogonal (angle between them 90 degrees) if: x dot y = 0 The Attempt at a Solution There's a similar example in my book, but...

Homework Statement If {u1, u2,...,um} are nonzero pairwise orthogonal vectors of a subspace W of dimension n, prove that m \leq n. The Attempt at a Solution I look at all my notes but I still can't understand what this qurstion asks or what definitions I need to be using for this... I'm...

My professor stated that the following orthogonality condition holds: \sum_{n=0}^N cos(2\pi mn/N)cos(2\pi kn/N)=0 where m != k, and 0<= m,k < N. I couldn't prove this, so I plugged in specific values: N=4, m=1, k=3. I found that the sum equals 2. Likewise for other situations where...

Homework Statement What is \int_0^{2 \pi} \; d\theta \sin^2 k\theta \cos^2 k\theta \; ? Homework Equations Orthogonality of sines and cosines? The Attempt at a Solution I tried substitution and didn't get anywhere. Yeah, that's about it.

i know that there could be the use of integrals in orthogonal things ??

Homework Statement A set of eigenfunctions yn(x) satisfies the Sturm-Liouville equation #1 with boundary conditions #2. The function g(x) = 0. Show that the derivatives un(x) = yn'(x) are also orthogonal functions. Determine the weighting function w(x) for these functions. What boundary...

Homework Statement Let u1 = [2 -2 2 ] u2 = [-2 2 1], u3 = [0 1 2]Use the Gram-Schmidt process to u1, u2, u3, in this order. The resulting vectors are: v1 = [___ ___ ___], v2 = [___ ___ ___], v3 = [___ ___ ___] And ß = {v1, v2, v3} is an orthongal basis for R3. Homework Equations v1 =...

Homework Statement Are all electronis states orthonormal? I mean the degenerate states ie [n,l,m>states corresponding to same energy for example can one write [2,0,0>=a[2,1,-1>+b[2,1,0>+c[2,1,+1> Homework Equations The Attempt at a Solution for example can one write...

Homework Statement With knowledge of the orthogonality conditions for eigenfunctions with discrete eigenvalues, determine the orthonormal set for eigenfunctions with continuous eigenvalues. Use the definition of completeness to show that | a(k) |^2 = 1. 2. The attempt at a solution The first...

I'm dealing with image transforms.These are of course 2D. I always thought orthogonality was the same as perpendicularity, so the max number of orthogonal bases you could come up with in 2D is 2. However, image processing is full of transforms such as Hadamard, Haar, etc. that can have...

I'm studying for my Quantum Computing exam. It's at 2 PM EST today. If anyone can give me a nudge in the right direction before then that would be excellent! Problem: Assume the operators P_i satisfy: \textbf{1} = \sum_i{P_i} P_i^{\dagger} = P_i P_j^2 = P_j. Show that P_i P_j = 0 whenever...

Homework Statement Show that: \varphi_{0}(x) = f_{0}(x) and \varphi_{1}(x) = f_{1}(x) - \frac{\left\langle\right\varphi_{0},f_{1}\rangle}{\left\|\varphi_{0}\right\|^{2}}\varphi_{0}(x) are orthogonal on the interval [a,b]. Homework Equations Orthogonal functions satisfy...

Maxwell's equations give that the electric and magnetic fields in E-M radiation are orthogonal. This is a classic equation, but can it be related to the orthogonality of, for example, the momentum and position operators which lead to non-commutivity?