What is Orthogonal: Definition and 581 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. Ventrella

    A Differences between Gaussian integers with norm 25

    I am exploring Gaussian integers in terms of roots, powers, primes, and composites. I understand that multiplying two integers with norm 5 result in an integer with norm 25. I get the impression that there are twelve unique integers with norm 25, and they come in two flavors: (1) Four of them...
  2. L

    Linear Algebra - Find an orthogonal matrix P

    A problem that I have to solve for my Linear Algebra course is the following We are supposed to use Mathematica. What I have done is that I first checked that A is symmetric, i.e. that ##A = A^T##. Which is obvious. Next I computed the eigenvalues for A. The characteristic polynomial is...
  3. P

    I Orthogonal matrix construction

    Given a real-valued matrix ## \bar{B}_2=\begin{bmatrix} \bar{B}_{21}\\ \bar{B}_{22} \end{bmatrix}\in{R^{p \times m}} ##, I am looking for an orthogonal transformation matrix i.e., ##T^{-1}=T^T\in{R^{p \times p}}## that satisfies: $$ \begin{bmatrix} T_{11}^T & T_{21}^T\\ T_{12}^T...
  4. Math Amateur

    MHB Orthogonal Projection .... .... D&K Example 1.5.3 .... ....

    I am reading "Multidimensional Real Analysis I: Differentiation" by J. J. Duistermaat and J. A. C. Kolk ... I am focused on Chapter 1: Continuity ... ... I need help with an aspect of Example 1.5.3 ... Duistermaat and Kolk"s Example 1.5.3 reads as follows:In the above example we read the...
  5. C

    I Is there a geometric interpretation of orthogonal functions?

    Hi all. So to start I'll say I'm just dealing with functions of a real variable. In my linear algebra courses one thing was drilled into my head: "Algebraic invariants are geometric objects" So with that in mind, is there any geometric connection between two orthoganal functions on some...
  6. G

    Orthogonal Projection of Perfect Fluid Energy Momentum

    Homework Statement Derive the relativistic Euler equation by contracting the conservation law $$\partial _\mu {T^{\mu \nu}} =0$$ with the projection tensor $${P^{\sigma}}_\nu = {\delta^{\sigma}}_\nu + U^{\sigma} U_{\nu}$$ for a perfect fluid. Homework Equations $$\partial _\mu {T^{\mu \nu}} =...
  7. C

    Orthogonal projection over an orthogonal subspace

    Homework Statement Being F = (1,1,-1), the orthogonal projection of (2,4,1) over the orthogonal subspace of F is: a) (1,2,3) b) (1/3, 7/3, 8/3) c) (1/3, 2/3, 8/3) d) (0,0,0) e) (1,1,1) The correct answer is B Homework Equations The Attempt at a Solution Using the orthogonal projection...
  8. S

    I Relationship between a non-Hermitian Hamiltonian and its solution

    Hello, I Have a non-Hermitian Hamiltonian, which is defined as an ill-condition numbered complex matrix, with non-orthogonal elements and linearily independent vectors spanning an open subspace. However, when accurate initial conditions are given to the ODE of the Hamiltoanian, it appears to...
  9. M

    MHB Adjoint operator and orthogonal projection

    Hello, I want to show $ T^{*}(Pr_{C}(y)) = Pr_{T^{*}(C)}(T^{*}y)$ where $T \in B(H)$ and $TT^{*}=I$ , $H$ is Hilbert space and $C$ is a closed convex non empty set. but i don't know how to start, or what tricks needed to solve this type of problems. also i want know how to construct $T$ to...
  10. R

    B 2 clocks -- Using orthogonal light path detectors in a space ship

    Hi I can't see the error in this can someone please explain where I went wrong? A man is in a spaceship traveling at a constant velocity He makes 2 identical tubes of length L with a mirror at one end, tube a and tube b He has a single light bulb. Next to the bulb is a detector. He carefully...
  11. Ron19932017

    I Self-Study GR: Construct Contravarient/Covarient Orthogonal Basis

    Hi everyone, I am trying to self study some general relativity however I met some problem in the contravarient and covarient basis. In the lecture, or you can also find it on wiki page 'curvilinear coordinates', the lecturer introduced the tangential vector ei =∂r/∂xi and the gradient vector ei...
  12. L

    MHB Finding a parameter for which a line is orthogonal to a curve

    Hiya again, I am trying to solve this problem, I thought I got somewhere, but kinda stuck. The graph of y^2=x^3 is called a semicubical parabola. Determine the constant b so that the line y = -(1/3)x+b meets this graph orthogonally. I found the derivative of the curve by using implicit...
  13. S

    I Is the Christoffel symbol orthogonal to the four-velocity?

    Consider a force-free particle moving on a geodesic with four-velocity v^\nu. The formula for the four-acceleration in any coordinate system is \frac{dx^\mu}{d\tau} = - \Gamma^\mu_{\nu\lambda} v^\nu v^\lambda Since the four-acceleration on the left side is orthogonal to the four-velocity, this...
  14. W

    QP: Physical Meaning of Orthogonality

    Homework Statement I have recently come across the notation <ψ|Φ> in my notes and am not quite sure what it means. Some articles I have read online state that this is analogous to the dot product, except that this is the "dot-product" of 2 wave-functions. Would I then be right in saying that...
  15. koustav

    Are Spacelike and Timelike Orthogonal: Mathematical Proof Explained

    are spacelike and timelike orthogonal?what is the mathematical proof
  16. ThunderLight

    Orthogonal Polarisation in EM waves and Interference

    I've been trying to get my head around Polarisation and how it achieves orthogonality. I'm not sure if this should be in Physics or Electrical Engineering Section. (Mods can move this where appropriate) I know that 2 EM wave with linear polarisations where one wave is shifted by π, they would...
  17. Minal

    A Find vectors in Orthogonal basis set spanning R4

    An orthogonal basis set spanning R4 has four vectors, v1, v2, v3 and v4. If v1 and v2 are [ −1 2 3 0 ] and [−1 1 −1 0 ] find v3 and v4. Please explain this in a very simple way.
  18. 0

    Eigenvectors and orthogonal basis

    Homework Statement I have a linear transformation ##\mathbb{R}^3 \rightarrow \mathbb{R}^3##. The part that asks for a basis of eigenvectors I've already solved it. The possible eigenvectors are ##(1,-3,0), (1,0,3), (\frac{1}{2}, \frac{1}{2},1) ##. Now the exercise wants me to show that there is...
  19. M

    Show GL/O/SO(n,R) form groups under Matrix Multiplication

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  20. M

    Prove all Elements of O(2,R) have form of Rotation Matrix

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  21. Vitani11

    Are two vectors that are orthogonal to a third parallel?

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  22. karush

    MHB S6.194.4.12.4.29 Find a nonzero vector orthogonal to plane

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  23. MarkFL

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  24. J

    MHB The Symmetry Group of a Hexagonal Prism and Orthogonal Matrices

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  25. Erenjaeger

    Finding orthogonal unit vector to a plane

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  26. Mr Davis 97

    I Orthogonal basis to find projection onto a subspace

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  27. Mr Davis 97

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  28. Kernul

    Line orthogonal to a plane with variable parameter

    Homework Statement I have to find for which values of the real parameter ##b## the following plane is orthogonal to the following line: ##\pi : 5x + (2b - 1)y - (1 + 8b)z + 3 = 0## ##s : \begin{cases} x + z - 4 = 0 \\ x - 3y + z + 2 \end{cases}## Homework EquationsThe Attempt at a Solution For...
  29. Kernul

    Checking if line is orthogonal to two skew lines

    Homework Statement So, I'm doing a long exercise, you can check here the first part: https://www.physicsforums.com/threads/checking-if-the-following-lines-are-coplanar.885948/ The second part asks me to find, if one of the couple of lines are skew, the orthogonal line to two skew lines...
  30. M

    Proving a matrix is orthogonal

    Homework Statement Show that the matrix ##P = \big{[} p_{ij} \big{]}## is orthogonal. Homework Equations ##P \vec{v} = \vec{v}'## where each vector is in ##\mathbb{R}^3## and ##P## is a ##3 \times 3## matrix. SO I guess ##P## is a transformation matrix taking ##\vec{v}## to ##\vec{v}'##. I...
  31. dreens

    I Orthogonal 3D Basis Functions in Spherical Coordinates

    I'd like to expand a 3D scalar function I'm working with, ##f(r,\theta,\phi)##, in an orthogonal spherical 3D basis set. For the angular component I intend to use spherical harmonics, but what should I do for the radial direction? Close to zero, ##f(r)\propto r##, and above a fuzzy threshold...
  32. entropy1

    B Can a Binary Number be Viewed as an Orthogonal Basis?

    Could you view a discrete number, for instance a binary number, as a sort of orthogonal basis, where each digit position represents a new dimension? I see similarities between a binary number and for instance Fourier Transform, with each digit being a discrete function.
  33. DavideGenoa

    I Lebesgue measure under orthogonal transofrmation

    Hello, friends! Let us define the external measure of the set ##A\subset \mathbb{R}^n## as $$\mu^{\ast}(A):=\inf_{A\subset \bigcup_k P_k}\sum_k m(P_k)$$where the infimum is extended to all the possible covers of ##A## by finite or countable families of ##n##-paralleliped ##P_k=\prod_{i=1}^n...
  34. K

    Orthogonal array for DOE by using the Taguchi method

    Hi in my project their are three variables like speed ,depth of cut and feed and each variable have 5 values then please tell me which Orthogonal array is suitable for my design of experiments by using taguchi method And tell me detailed Thank you
  35. terryds

    Orthogonal vector condition

    Homework Statement Vector u, v, and x are not zero. Vector u + v will be perpendicular (orthogonal) to u-x if A. |u+v| = |u-v| B. |v| = |x| C. u ⋅ u = v ⋅ v, v = -x D. u ⋅ u = v ⋅ v, v = x E. u ⋅ u = v ⋅ v Homework Equations u⋅v = |u||v| cos θ The Attempt at a Solution [/B] Two vectors are...
  36. Y

    Linear Algebra - Find Orthogonal Matrix Q that diagonals

    Homework Statement I'm told to find the matrix Q of the matrix A Homework EquationsThe Attempt at a Solution So my problem is that in the answer key they have S = (1/3)... and I have no idea where this 1/3 comes from. I get an equivalent answer for X_1, X_2, and X_3 S = [X_1, X_2, X_3] but...
  37. E

    I What is the scale factor in orthogonal vector calculus?

    Could someone explain to me in simplest of terms what scale factor is when dealing with orthogonal vectors.
  38. erbilsilik

    Research Orthogonal Lie Group for Physics Applications

    [Mentor's Note: Thread moved from homework forums] Where can I start to research this question? I did not take any course on Group theory before and I know almost nothing about the relationship with this pure maths and physics. I've decided to start with Arfken's book but I'm not sure. 1...
  39. T

    Finding the Orthogonal Complement

    Homework Statement Let V be the 2-dim subspace of R^3 spanned by V1 = (1,1,1) and V2 = (-2,0,1). Find the orthogonal compliment Vperpendicular. 2. Relevant equation X + Y + Z = 0 -5X + Y + 4Z = 0 The Attempt at a Solution Firstly, I orthogonalize the basis for V and get the vectors (1,1,1)...
  40. Pull and Twist

    MHB HELP: Orthogonal Trajectories

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  41. H

    Prove principal axes of moment of inertia are orthogonal

    What does the first two equations mean? I can't make sense of the notations. Does it mean taking the x-axis to be parallel to one principal axis and the y-axis to be parallel to the other principal axis? Source: http://hepweb.ucsd.edu/ph110b/110b_notes/node29.html EDIT: I figured it out. They...
  42. E

    I Eigenspectra and Empirical Orthogonal Functions

    Are the Eigenspectra (a spectrum of eigenvalues) and the Empirical Orthogonal Functions (EOFs) the same? I have known that both can be calculated through the Singular Value Decomposition (SVD) method. Thank you in advance.
  43. A

    I Are the derivatives of eigenfunctions orthogonal?

    We know that modes of vibration of an Euler-Bernoulli beam are given by eigenfunctions, with the natural frequency of each mode being given by its eigenvalue. Thus these modes are all mutually orthogonal.Can anything be said of the derivatives of these eigenfunctions? For example, I have the...
  44. RJLiberator

    PDE: Proving that a set is an orthogonal bases for L2

    Homework Statement Show that the set {sin(nx)} from n=1 to n=∞ is orthogonal bases for L^2(0, π). Homework EquationsThe Attempt at a Solution Proof: Let f(x)= sin(nx), consider scalar product in L^2(0, π) (ƒ_n , ƒ_m) = \int_{0}^π ƒ_n (x) ƒ_m (x) \, dx = \int_{0}^π sin(nx)sin(mx) \, dx =...
  45. Steve Turchin

    Is this complex vector orthogonal to itself?

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  46. C

    Can an orthogonal matrix be complex?

    Can an orthogonal matrix involve complex/imaginary values?
  47. F

    How to prove that H_a and H_b are orthogonal?

    1. Okay, so I am going to prove that \int H_a\cdot H_bdv=0 Hint: Use vector Identities H is the Magnetic Field and v is the volume. Homework Equations this this[/B] k_bH_b=\nabla \times E_b k_aH_a=\nabla \times E_a k is the wave vector and E is the electric field The Attempt at a...
  48. E

    Cloning orthogonal quantum states - circuit?

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  49. ognik

    MHB Determinant of orthogonal 2 X 2

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  50. S

    Orthogonal matrices form a group

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