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.

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  1. DavideGenoa

    Orthogonality of Hermite functions

    Hi, friends! I want to show that Hermite functions, defined by ##\varphi_n(x)=(-1)^n e^{x^2/2}\frac{d^n e^{-x^2}}{dx^n}##, ##n\in\mathbb{N}## are an orthogonal system, i.e. that, for any ##m\ne n##, ##\int_{-\infty}^\infty e^{x^2} \frac{d^m e^{-x^2}}{dx^m} \frac{d^n e^{-x^2}}{dx^n}=0 ## I have...
  2. J

    Duality and Orthogonality: What's the Difference?

    1. I cannot understand the difference between orthogonality and duality? Of course orthogonal vectors have dot product zero but isn't this the condition of duality as well? Swinging my head around it my cannot find the answer on the internet as well. 2.Relating to same concept is orthogonality...
  3. P

    Orthogonality of wave functions

    Homework Statement Homework EquationsThe Attempt at a Solution http://i.imgur.com/tktQBsp.jpg [/B] I assume that you need to prove that the integral of psi1*psi0 is 0, so I have written out the integral and attempted to solve using integration by parts, but whichever way I write out the...
  4. G

    Proof of Trace Orthogonality Relation for Matrices $\Gamma^A$

    I know that the matrices {\Gamma^{A}} obey the trace orthogonality relation Tr(\Gamma^{A}\Gamma_{B})=2^{m}\delta^{A}_{B} In order to show that a matrix M can be expanded in the basis \Gamma^{A} in the following way M=\sum_{A}m_{A}\Gamma^{A} m_{A}=\frac{1}{2^{m}}Tr(M\Gamma_{A}) is it enough to...
  5. JohnPrior3

    Subspace Orthogonality in Ax=b

    Let A be the matrix [2 0 1 0 1 -1 4 3 3 -1 5 3] Let b= [b1 b2 b3] transpose What equation must be satisfied by the components of b in order to guarantee that there will exists a vector x= [x1 x2 x3 x4] transpose satisfying the equation Ax=b. Justify your answer. I know C(A) is the orthogonal...
  6. F

    Proving Orthogonality by Induction in Second Quantization for Bosonic Systems

    Hello, I'm currently studying second quantization. I need to prove <n^\prime| n>=\delta_{n^\prime n} by mathematical induction in the number of particles representation. However I don't know how to do this proof having two natural numbers n and n^\prime. Must I prove it holds for <0|0>, <0|1>...
  7. 1

    Orthogonality and physical applications

    Just as we have orthogonal vectors/vector spaces/etc., we can have orthogonal functions/function spaces/etc. I'm trying to apply these concepts to physical processes. Here's a general idea of what I'm doing: Suppose you have a physical quantity you are trying to measure, ##F##, and it depends...
  8. L

    Static + Hypersurface Orthogonality

    Static spacetimes can be defined as having no g_{tx} component of the metric. Alternatively we can say that they are foliated by a bunch of spacelike hypersurfaces to which the Killing vector field \frac{\partial}{\partial t} is orthogonal. How are these two statements consistent...
  9. skujesco2014

    Orthogonality condition for Airy functions

    Hi, all. I'll be brief. Can Airy functions [those who solve the differential equation y''-xy=0] be considered orthogonal over some interval? If so, what is their orthogonality condition? Given that the Airy functions have a representation in terms of Bessel functions, I would be inclined to...
  10. M

    Fermi Surface and Orthogonality Catastrophe

    I am reading section 8.5.1 of http://f3.tiera.ru/2/P_Physics/PS_Solid%20state/Giuliani%20G.,%20Vignale%20G.%20Quantum%20theory%20of%20the%20electron%20liquid%20%28CUP,%202005%29%28ISBN%200521821126%29%28799s%29_PS_.pdf (page 442 of the book, page 465 of the pdf). The author claims the...
  11. T

    Orthogonality of sine and cosine question

    Hello, I'm trying to solve Fourier Series, but I have a question. I know that cos(nx) is even and sin(nx) is odd. But what does this mean when I take the integral or sum of cos(nx) or sin(nx)? Do they have a value or do they just keep their form?
  12. D

    Proving Orthogonality of θ_{ik}V^k to U_i

    Homework Statement Show that the tensor θ_{ik} = g_{ik} - U_{i}U_{k} projects any vector, V^{k}, into a 3-surface orthogonal to the unit time-like vector U_{i} (By a projection, the vector θ_{ik}V_{k}, is implied). Homework Equations The Attempt at a Solution The projection should be...
  13. F

    Integration by parts with orthogonality relation

    Homework Statement I want to integrate \int_{0}^{a} xsin\frac{\pi x}{a}sin\frac{\pi x}{a}dxHomework Equations I have the orthogonality relation: \int_{0}^{a} sin\frac{n\pi x}{a}sin\frac{m\pi x}{a}dx = \begin{cases} \frac{a}{2} &\mbox{if } n = m; \\ 0 & \mbox{otherwise.} \end{cases} and...
  14. M

    Linear Algebra orthogonality problem

    Homework Statement Let W be the intersection of the two planes x + y + z = 0 and x - y + z = 0 In R3. Find an equation for Wτ Homework Equations The Attempt at a Solution So, W = {(x, y, z) l 2y =0} I don't think that is a correct was to represent W being...
  15. L

    Linear independancy and orthogonality of vectors

    Hi, I'm reading up on linear algebra and I'm wondering if the remark after a theorem I'm reading here is complete. The theorem states: "If {V_1,V_2,...,V_k} is an orthogonal set of nonzero vectors then these vectors are linearly independent." Remark after that simply states that if a set of...
  16. 0

    Orthogonality of wave functions to negative momentum states

    This is a question I have about the textbook discussion, so I'll do away with the standard format. The author of my QM book (Shankar, Principles of Quantum Mechanics) used the term "negative momentum states," all of a sudden, and I've never heard of it before. He has a little note saying that...
  17. P

    Examples of Diminnie orthogonality

    Definition of this orthogonality goes like this: ## x, y \in X##, where ##X## - normed space and ##X^*## - its dual space. Then ##x## is orthogonal ##y##, if $$ \sup\{f(x)g(y)-f(y)g(x)|, \, f,g\in X^*, \|f\|,\|g\|≤1\}=\|x\|\|y\| $$ From what I understand ##f## and ##g## are linear...
  18. D

    Hydrogen atom eigenstate: proving orthogonality of states

    Homework Statement Hey dudes So here's the question: Consider the first excited Hydrogen atom eigenstate eigenstate \psi_{2,1,1}=R_{2,1}(r)Y_{11}(\theta, \phi) with Y_{11}≈e^{i\phi}sin(\theta). You may assume that Y_{11} is correctly normalized. (a)Show that \psi_{2,1,1} is orthogonal...
  19. J

    How to prove orthogonality on a set of functions?

    Homework Statement A set of functions, F, is given below. Determine the size of the largest subset of F which is mutually orthogonal on the interval [-1, 1], and find all such subsets of this size. Show all of your work. F = { 1, x, x2 , sin(x), cos(x), cosh(x), sinh(x)}Homework Equations Not...
  20. nomather1471

    Orthogonality of timelike and null vector

    Can we show orthogonality of timelike and null vector?
  21. E

    Trigonometric Orthogonality Query

    Hello everyone, I've wandered PF a few times in the past but never thought I'd join, here I am, how exciting. To keep it short I'm trying to understand the proof behind Fourier Series and can't quite get to grips with basic trigonometric orthogonality. I understand that sin and cos are...
  22. I

    Orthogonality condition for disimilar Bessel functions

    As per orthogonality condition this equation is valid: \int_0^b xJ_0(\lambda_nx)J_0(\lambda_mx)dx = 0 for m\not=n I want to know the outcome of the following: \int_0^b xJ_0(\lambda_nx)Y_0(\lambda_mx)dx = 0 for two cases: m\not=n m=n
  23. I

    Application of orthogonality condition

    Homework Statement I have applied separation of variables to a transient radial heat equation problem. T is a function of r and t. I have reached the following step: Homework Equations T_2(t,r) = \sum_{m=1}^ \infty c_m...
  24. Y

    Orthogonality and orthogonal set.

    I am brushing up this topic. I want to verify both orthogonality between two functions and an orthogonal set ALWAYS have to be with respect to the specified interval...[a,b]. That is, a set of {1, ##\cos n\theta##, ##\sin m\theta##} is an orthogonal set IF AND ONLY IF ##\theta## on...
  25. F

    Coefficients using orthogonality relation

    Homework Statement Have a solution for the temperature u(x,t) of a heated rod, now using the orthogonality relation below show that the coefficients a_n , n = 0,1,2,... can be expressed as: a_n = \frac{2}{L} \int_{0}^{L} cos\frac{n\pi x}{L} f(x) dx Homework Equations \int_{0}^{L}...
  26. G

    Orthogonality Problem (From Fourier Analysis Text)

    Hello all, I'm working through a majority of the problems in "A First Course in Wavelets with Fourier Analysis" and have stumbled upon a problem I'm having difficulty with. Please view the PDF attachment, it shows the problem and what I have done with it so far. Once you have seen the...
  27. D

    Orthogonality functions

    Homework Statement Prove that the eigenfunctions of (1) are orthogonal. m y_{tt} + 2Uy_{tx} + U^2y_{xx} +EIy_{xxxx} =0 (1) for 0<x<L, t>0 with y(0,t) = 0 y_x(x,t) = 0 y_{xx}(L,t) = 0 y_{xxx}(L,t) = 0 y(x,0) = f(x) y_t (x,0) = g(x) The...
  28. P

    Understanding the Physical Meaning of Orthogonality Condition in Functions

    What does it mean when we say that two functions are orthogonal (the physical meaning, not the mathematical one)? I tried to search for the physical meaning and from what I read, it means that the two states are mutually exclusive. Can anyone elaborate more on this? Why do we impose...
  29. Fernando Revilla

    MHB C.'s question at Yahoo Answers (orthogonality).

    Here is the question: Here is a link to the question: Can someone help me with this linear algebra question (Orthogonality) ? - Yahoo! Answers I have posted a link there to this topic so the OP may find my response.
  30. Y

    Lineaer algebra - orthogonality

    Homework Statement Let<x,y> be an inner product on a vector space V, and let e1, e2,...,en be an orthonormal basis for V. Prove: <x,y> = <x,e1><y,e1>+...+<x,en><y,en>. Homework Equations <x,x> = abs(x) a<x,y> = <ax,y> = <x,ay> The Attempt at a Solution RHS =<x,y> =...
  31. S

    On Bessel function's orthogonality

    Use the orthogonality relation of Bessel function to argue whether the following two integrals are zero or not: \displaystyle\int_0^1J_1(x)xJ_2(x)dx \displaystyle\int_0^1J_1(k_1x)J_1(k_2x)dx, where k_1,k_2 are two distinct zeros of Bessel function of order 1. The textbook we are using is...
  32. M

    Proof of orthogonality of harmonic conjugates

    Homework Statement My textbook (Churchill) is asking me to prove that the contours $$u(x,y) = c_1$$ and $$v(x, y) = c_2$$ where $$u$$ and $$v$$ are the real and imaginary components of an analytic function $$f(z)$$ are orthogonal at any point by noting that $$u_x + u_y \frac{dy}{dx} = 0 $$ and...
  33. R

    Inner Product Space and Orthogonality proof question (is this the correct way?)

    Homework Statement Let V be the inner product space. Show that if w is orthogonal to each of the vectors u1, u2,...,ur, then it is orthogonal to every vector in the span{u1,u2,...,ur}. Homework Equations u.v=0 to be orthogonal If u and v are vectors in an inner product space, then...
  34. J

    Orthogonality and Weighting Function of Sturm-Liouville Equation

    Homework Statement A set of eigenfunctions yn(x) satisfies the following Sturm-Liouville equation: \frac{d(f(x)*y'_{m})}{dx}+\lambda*\omega*y_{m}=0 with following boundary conditions: \alpha_{1}y+\beta_{1}y'=0 at x=a \alpha_{2}y+\beta_{2}y'=0 at x=b Show that the derivatives un(x)=y'n(x) are...
  35. dexterdev

    A doubt on stastical indeependence , orthogonality and uncorrelatedness ?

    A doubt on statistical independence , orthogonality and uncorrelatedness ? Hi friends, I wanted to make my concepts on statistical independence, uncorrelatedness and orthogonality clear. Suppose I have 2 random variables x and y. I have 2 pictures on the above concepts, is it...
  36. R

    Sturm-Liouville Question on Orthogonality

    Homework Statement For the following diff. eqns (fcns of t) X''m + λmXm=0 Xm (1)=0 X'm=0 X''n + λnXn=0 Xn (1)=0 X'n=0 Show that ∫XmXndt from 0 to 1 equals 0 for m≠n. Homework Equations Qualitative differential equations... no idea really what to put in this section. The...
  37. R

    Integral Product of Cosines to Show Orthogonality

    Homework Statement Show that (forgive me for not knowing how to use latex) from x=0 to x=1 of: ∫cos([(2n+1)(pi)/2]x)*cos([(2m+1)(pi)/2)]x) dx = 0, for m ≠ n Homework Equations The question tells me to use integral tables. The Attempt at a Solution Using integral tables, I got...
  38. D

    MHB Orthogonality of Eigenfunctions of Mixed Boundary Conditions

    $$ \left.(\phi_n\phi_m' - \phi_m\phi_n')\right|_0^L + (\lambda_m^2 - \lambda_n^2)\int_0^L\phi_n\phi_m dx = 0 $$ where $\phi_{n,m}$ and $\lambda_{n,m}$ represent distinct modal eigenfunctions which satisfy mixed boundary conditions at $x = 0,L$ of the form \begin{alignat*}{3} a\phi(0) + b\phi'(0)...
  39. 1

    Are ||b||a + ||a||b and ||b||a - ||a||b always orthogonal?

    Homework Statement Show that ||b||a + ||a||b and ||b||a - ||a||b are orthogonal vectors. Homework Equations The Attempt at a Solution After analyzing it and trying to prove it to no avail, I don't even think it's a true statement.
  40. R

    Orthogonality of Absolute Derivative

    Hello! Is any covariant vector orthogonal to absolute derivative of its contravariant counterpart? I read a GR book, and it says the tangent vector of a curve is orthogonal to its absolute derivative, that is ##D\lambda^A/dst_A=0##, where ##t^A## is the unit tangent vector of some curve...
  41. P

    Proving Orthogonality of $\hat{A}|\Psi>$ with Anti-Unitary Operator $\hat{A}$

    Homework Statement \hat{A} is an anti-unitary operator, and it is known that \hat{A^2}= -\hat{I}, show that |\Psi> is orthogonal to \hat{A}|\Psi> Homework Equations I know that \hat{A} can be represented by a unitary operator, \hat{U}, and the complex conjugation operator, \hat{K}...
  42. A

    About orthogonality and nullspaces

    Hi: I see an example about nullspace and orthogonality, the example is following: $$Ax=\begin{bmatrix} 1 & 3 &4\\ 5 & 2& 7 \end{bmatrix} \times \left[ \begin{array}{c} 1 \\ 1\\-1 \end{array} \right]=\begin{bmatrix} 0\\0\end{bmatrix}$$ The conclusion says the nullspace of A^T is only the zero...
  43. B

    MHB Orthogonality of stirling numbers

    Dear MHB members, denote by $s_{n,k}$ and $S_{n,k}$ Stirling numbers of the first-kind and of the second-kind, respectively. I need to see the proof of the identity $\sum_{j=k}^{n}S(n,j)s(j,k)=\sum_{j=k}^{n}s(n,j)S(j,k)=\delta _{{n,k}}$. Please let me know if you know a reference in this...
  44. A

    Complex orthogonality of electric and magnetic fields

    Hi, I have a hard time finding a justification that electric and magnetic fields are still orthogonal when presented in complex form. As far as I know the notion of orthogonality for complex vectors is not as intuitive as the one for real vectors. Notably, \vec{x}\cdot\vec{y}=0 does not imply...
  45. M

    Orthogonality Relationship for Legendre Polynomials in Cylindrical Coordinates

    Hello everyone, Sorry if this is in the wrong sub-forum, I wasn't sure exactly where to place it. I was wondering if there is an orthogonality relationship for the Legendre polynomials P^{0}_{n}(x) that have been converted to cylindrical coordinates from spherical coordinates, similar to...
  46. D

    Easy method to show orthogonality of a matrix

    Hi, I know how the properties of an orthogonal matrix, the transpose ot the matrix is equal to its inverse. The problem is that the teacher gaves me a 3x3 matrix expressed in terms of many cosines and sines of three angles, I want to know how can I prove that the matrix is orthogonal without...
  47. S

    Intersect of U and U perpendicular; Orthogonality

    Homework Statement Let U be a subspace of ℝn. Show that if u\inU\bigcapU\bot, then u=0. Homework Equations The Attempt at a Solution I know that U\bot will be orthogonal to U, so any vector u in U dotted with any vector in U\bot will equal 0. But that does not necessarily mean...
  48. A

    Are all eigenstates of observables orthogonal?

    Suppose psi1 and psi2 are eigenstates of observables O1 and O2 Suppose Value of O1 of psi1 = value of O1 of psi2 Therefore, <psi1|psi2>=1 Suppose value of O2 of psi1<>value of O2 of psi2 Therefore <psi1|psi2>=0 Contradiction!how to explain
  49. Sigurdsson

    Orthogonality of the coefficients of the Lorentz transformation

    Homework Statement For flat spacetime the coefficients of the Lorentz transformation are defined as \alpha^{\nu}_{\mu} = \frac{\partial x^{' \nu}}{\partial x^{\mu}} Whereas the Lorentz transformation is \begin{pmatrix} x_1' \\ x_2' \\ x_3' \\ x_4' \end{pmatrix} = \begin{pmatrix}...
  50. Z

    Orthogonality of cosine and sine functions

    Can someone give a more intuitive explanation on how it is (if it is true), that; ∫all cos (nx) cos (mx) = 0 if n!=m or ∫all sin (nx) sin (mx) = 0 if n!=m thanks
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