What is Orthogonality: Definition and 175 Discussions

In mathematics, orthogonality is the generalization of the notion of perpendicularity to the linear algebra of bilinear forms. Two elements u and v of a vector space with bilinear form B are orthogonal when B(u, v) = 0. Depending on the bilinear form, the vector space may contain nonzero self-orthogonal vectors. In the case of function spaces, families of orthogonal functions are used to form a basis.
By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in other fields including art and chemistry.

View More On Wikipedia.org
Recent contents Top users
  1. Demon117

    Orthogonality of Legendre Polynomials from Jackson

    Hello all! I am trying to work through and understand the derivation of the Legendre Polynomials from Jackson's Classical electrodynamics. I have reached a part that I cannot get through however. Jackson starts with the following orthogonality statement and jumps (as it seems) in his proof...
  2. M

    Linear Algebra: Dot Product and Orthogonality

    This is from my homework, I was moving along nicely until I hit this problem, (there's another just like it right after this). I can't find reference for solving this in the chapter I am looking at. The answer is in the back of the book….-2911. Can someone explain this to me? ||\mathbf{u}|| =...
  3. J

    Orthogonality in h^n for a field h

    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...
  4. S

    Linearity and Orthogonality of Inner Product in Vector Space H

    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...
  5. P

    Dot/bilinear product in C^n / Orthogonality

    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...
  6. W

    Exploring the Orthogonality of Sine and Cosine Functions in Fourier Series

    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...
  7. W

    Orthogonality in Legendre polynomials

    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...
  8. S

    Orthogonality theorem proof method question

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

    Associated Legendre functions and orthogonality

    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...
  10. G

    Eigenvector orthogonality and unitary operator diagonalization

    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...
  11. A

    A question on orthogonality relating to fourier analysis and also solutions of PDES

    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...
  12. S

    Givens Rotation: Find J(2,3) and Prove Orthogonality

    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...
  13. S

    Exploring the Wedge Product & its Role in Vectors & Orthogonality

    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...
  14. H

    Fundamental theorem of Orthogonality

    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...
  15. N

    Orthogonality and orthonormality ?

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

    The fundamental orthogonality theorem

    Homework Statement Hello everyone! I am trying to do my homework and I wonder if any of you knows what ''the fundamental orthogonality theorem'' is...? (my teacher calls it like that) I have googled it but the only thing I could find was orthogonality... I could not find something like ''the...
  17. P

    Orthogonality limits of Bessel Polynomials

    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...
  18. J

    Normalization and orthogonality of wavefunctions

    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...
  19. B

    What is The usage of orthogonality & orthonormal in useful life

    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?
  20. T

    Orthogonality, Fourier series and Kronecker delta

    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...
  21. X

    Orthogonality of Legendre Polynomials

    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...
  22. M

    Prove Orthogonality Condition For Sines (Integral)

    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...
  23. L

    Orthogonality of time dependent vector derivatives of constant magnitude

    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.
  24. G

    Orthogonality- Gram-Schmidt Process for Complex Sequences

    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 =...
  25. P

    Bessel Function, Orthogonality and More

    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...
  26. T

    Orthogonality between optical fibre modes

    Hi there, I've just read the following: The expression that is given is: \int_{A \infty} e_j \times h_k* \cdot \widehat{z} dA = 0 where * denotes the complex conjugate, and z^ is the unit vector in the direction of propagation (along the axis of the fibre). Can anyone explain...
  27. K

    Subspaces and Orthogonality

    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...
  28. P

    Orthogonality integral help

    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...
  29. mnb96

    Are Complex Sinusoids an Orthogonal Basis for L^2(\mathbb{R}) Space?

    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...
  30. mnb96

    Are Shifted Sinc Functions Orthogonal for Any Real Value of x_0?

    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}\{...
  31. H

    Understanding Orthogonality in Wavefunctions

    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...
  32. J

    Using orthogonality properties

    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...
  33. M

    Proof of orthogonality of associated Legendre polynomial

    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...
  34. X

    Orthogonality of wave function of finite potential well

    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...
  35. L

    Orthogonality of Matsubara Plane Waves

    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...
  36. H

    Orthogonality of Sine and Cosine functions

    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
  37. M

    Proving Vector Orthogonality in R^3

    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...
  38. C

    Orthogonality and find coefficients

    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...
  39. R

    Calculating Orthogonality of Binormal Vector with Dot Product

    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?
  40. A

    Orthogonality Property of Hyperbolic functions ?

    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...
  41. M

    Vector Planes & Orthogonality - Help

    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...
  42. S

    Orthogonality, point on line closest to point in space

    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...
  43. R

    Dimension and Orthogonality in Vector Spaces: A Proof of the Inequality m ≤ n

    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...
  44. B

    Orthogonality in Discrete Fourier Transforms

    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...
  45. B

    Exploring Orthogonality: A Homework Challenge

    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.
  46. S

    Legendre Polynomial Orthogonality Integral Limits

    Good afternoon I have a question regarding the limits on the orthogonality integral of Legendre Polynomicals: \int_{-1}^1 P_l(u)P_{l'}du = 2/(2l+1) I am in the middle of a question involving the solution of Laplace's equation inside a hemisphere, which means that for the usual...
  47. T

    Looking for a question in orthogonality

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

    Sturm-Liouville Orthogonality Proof

    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...
  49. H

    Vectors Question (Orthogonality)

    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 =...
  50. Q

    Can Degenerate States be Expressed as a Linear Combination of Orthogonal States?

    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...
Back
Top