What is Orthogonal: Definition and 582 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. A

    HCP miller indices in Orthogonal coordinate system

    Hi everyone, I am doing MD simulation for zirconium (hcp). I have to input some orientation for crystal in simulation. But i have orientation in 4-index bravais miller indices. and i have to convert (plane and direction) it from 4-index to 3-index orthogonal coordinate system. Please help me...
  2. J

    Write F as a sum of an orthogonal and parallel vector

    an object is moving in the direction i + j is being acted upon by the force vector 2i + j, express this force as the sum of a force in the direction of motion and a force perpendicular to the direction of motion. the parallel would be \hat{i}+\hat{j} and the orthogonal would be \hat{i} -...
  3. D

    Orthogonal operator and reflection

    Homework Statement Let ##n## be a unit vector in ##V## . Define a linear operator ##F_n## on ##V## such that $$F_n(u) = u-2\langle u, n \rangle n \; \mathrm{for} \; u \in V.$$ ##F_n## is called the reflection on ##V## along the direction of ##n##. Let ##S## be an orthogonal linear operator on...
  4. F

    Angle between two orthogonal coordinate systems

    If two orthogonal coordinate systems (xyz and x'y'z') share a common origin, and the angles between x and x', y and y', and z and z' are known. What is angle between the projection of z' on the xy plane and the x axis? Thank you for your help!
  5. M

    MHB The eigenvalues are real and that the eigenfunctions are orthogonal

    Hey! :o We have the Sturm-Liouville problem $\displaystyle{Lu=\lambda u}$. I am looking at the following proof that the eigenvalues are real and that the eigenfunctions are orthogonal and I have some questions... $\displaystyle{Lu_i=\lambda_iu_i}$ $\displaystyle{Lu_j=\lambda_ju_j...
  6. N

    Learn Legendre's Polynomial & Orthogonal Functions Quickly

    Can anyone tell me where can I learn about Legendre's Polynomial and Orthogonal functions in a jiffy? I've to use it to solve my scattering problem. I know a thing or two about it but don't know how to use in a specific problem. https://www.physicsforums.com/showthread.php?t=410830
  7. M

    Orthogonal complement of even functions

    Homework Statement . Let the ##\mathbb R##- vector space ##C([-1,1])=\{f:[-1,1] \to \mathbb R\ : f \space \text{is continuous}\ \}## with the inner product ##<f,g>=\int_{-1}^1 f(t)g(t)dt##. Determine the orthogonal complement of the subspace of even functions (call that subspace ##S##). The...
  8. M

    Is a Vector Orthogonal to a Set Also Orthogonal to Its Span?

    Homework Statement Let V be an 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 The Attempt at a Solution Not sure how to show this, if w is orthogonal to...
  9. jk22

    What happens with non orthogonal eigenvectors

    I considered the covariance of 2 spin 1/2 as a non linear operator : A\otimes B-A|\Psi\rangle\langle\Psi|B. The eigenvectors are but non orthogonal and I wondered what happens in that case with the probabilities : from Born"s rule it comes that the transition probability from one vector to the...
  10. C

    Solving for Orthogonal Vectors in R4?

    Homework Statement Given following vectors in R4: v= (4,-9,-6,3) u = (5,-8,k,4) w=(s,-5,4,t) A. Find value of k if u and v are orthogonal B. Find values of S and T if w and u are orthogonal and w and v are orthogonal Homework Equations Orthogonal means dot product is zero...
  11. N

    Nullspace of Orthogonal Vectors: Example Matrix

    Homework Statement Let S be the subspace of all vectors in R4 that are orthogonal to each of the vectors (0, 4, 4, 2), (3, 4, -2, -4) What is an example of a matrix for which S is the nullspace? The Attempt at a Solution I'm not sure how I should be intepreting the question: [ 0 ,4 ,4 ,2...
  12. AwesomeTrains

    Finding the Orthogonal Complement of a Vector Space

    Homework Statement G:= \begin{pmatrix} 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 0 \\ 0 & 1 & 1 & 0\\ 1 & 0 & 0 & 1 \\ \end{pmatrix} B(x,y) = x^{T}Gy B: \textit{R}^{4} X \textit{R}^{4} \rightarrow \textit{R} Find (\textit{R}^{4})^{\bot} Homework Equations...
  13. V

    Volume Integral Orthogonal Polynomials

    Hello. Homework Statement Basically I want to evaluate the integral as shown in this document: Homework Equations The Attempt at a Solution The integral with the complex exponentials yields a Kronecker Delta. My question is whether this Delta can be taken inside the integral...
  14. F

    Orthogonal space-like vectors.

    Hey, I read that if a four-vector is 'four-orthogonal' to a time-like four vector then it must be space-like. I showed this quite easily. I also read that a space-like vector can be orthogonal to another space-like vector, but can't seem to prove it. I wondered if someone could help me.
  15. J

    Angle between U and V when given perpendicular vectors A and B

    Homework Statement vector A = 3U-V vector B = U+2V U and V are vectors |U| = 3|V| Given that vector A and vector B are perpendicular vectors, find the angle between vector U and vector V. Homework Equations A*B = |A||B|cos(∠AB) A*A = |A|^2 The Attempt at a Solution Since A and B are...
  16. C

    Prove the W is the orthogonal complement of its orthogonal complement

    Homework Statement Let W be a subspace of R^n. Show that the orthogonal complement of the orthogonal complement of W is W. i.e. Show that (W^{\perp})^{\perp}=W The Attempt at a Solution This is one of those 'obvious' properties that probably has a really simple proof but which...
  17. A

    Wave function orthogonal components

    1. The photon wave function, an EM wave, has orthogonal electric and magnetic components. I have gathered the impression that the electron wave function has only one. Is this correct? 2. By analogy with EM waves, can the electron's spin rate be identified with the frequency of its wave...
  18. ShayanJ

    How Do Different Coordinate Systems Affect Vector Operations?

    Recently I've been studying about orthogonal coordinate systems and vector operations in different coordinate systems.In my studies,I realized there are some inconsistencies between different sources which I can't resolve. For example in Arfken,it is said that the determinant definition of the...
  19. S

    Finding Orthogonal Vectors in R4 with Norm 1

    Homework Statement Find two vectors in R4 of norm 1 that are orthogonal to the vectors u = (2, 1, −4, 0), v = (−1, −1, 2, 2) and w = (3, 2, 5, 4). Homework Equations The Attempt at a Solution What i did was, i let a vector x = (x1, x2, x3, x4) that has a norm of 1 and...
  20. R

    Orthogonal Projections vs Non-orthogonal projections?

    Hi everyone, My Linear Algebra Professor recently had a lecture on Orthogonal projections. Say for example, we are given the vectors: y = [3, -1, 1, 13], v1 = [1, -2, -1, 2] and v2 = [-4, 1, 0, 3] To find the projection of y, we first check is the set v1 and v2 are orthogonal...
  21. Sudharaka

    MHB Invariant Orthogonal Compliment

    Hi everyone, :) Here's a question with my answer. I would be really grateful if somebody could confirm whether my answer is correct. :) Problem: Prove that the orthogonal compliment \(U^\perp\) to an invariant subspace \(U\) with respect to a Hermitian transformation is itself invariant...
  22. U

    Orthogonal projecitons, minimizing difference

    Homework Statement Determine the polynomial p of degree at most 1 that minimizes \int_0^2 |e^x - p(x)|^2 dx Hint: First find an orthogonal basis for a suitably chosen space of polynomials of degree at most 1 The Attempt at a Solution I assumed what I wanted was a p(x) of the form...
  23. evinda

    MHB Orthogonal trajectories of curves

    Hello ! I have to find the orthogonal trajectories of the curves : x^{2}-y^{2}=c , x^{2}+y^{2}+2cy=1 .. How can I do this?? For this: x^{2}-y^{2}=c I found \left | y \right |=\frac{M}{\left | x \right |} ,and for this: x^{2}+y^{2}+2cy=1 ,I found: y=Ax-D,c,A,D \varepsilon \Re ...
  24. Omega0

    Simple Integration Orthogonal Sin

    Hi, take the function I(m,n) = Integral from 0 to 1 of sin(m*pi*x)*sin(n*pi*x) over dx depending from n and m, being +-1, +-2, and so on. If I use sin(x)sin(y)=1/2(cos(x-y)-cos(x+y)) I get sin^2(n*pi*x)=1/2-1/2cos(2n*pi*x) or I(n,n)=1/2 because the integral of cosine over full...
  25. Sudharaka

    MHB Orthogonal Transformation in Euclidean Space

    Hi everyone, :) Here's one of the questions that I encountered recently along with my answer. Let me know if you see any mistakes. I would really appreciate any comments, shorter methods etc. :) Problem: Let \(u,\,v\) be two vectors in a Euclidean space \(V\) such that \(|u|=|v|\). Prove that...
  26. B

    Normalization of Orthogonal Polynomials?

    The generalized Rodrigues formula is of the form K_n\frac{1}{w}(\frac{d}{dx})^n(wp^n) The constant K_n is seemingly chosen completely arbitrarily, & I really need to be able to figure out a quick way to derive whether it should be K_n = \tfrac{(-1)^n}{2^nn!} in the case of Jacobi...
  27. A

    Orthogonal Matrices: Definition & Examples

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  28. D

    Value of combinaison of orthogonal elements

    Hello this is not a homework, just studying for the exam, and : Homework Statement Consider E a linear space with dot product (.,.) and the norm ||x|| = sqrt(x,x) a and b two orthogonals elements of E Find the value of ||a+ b|| et||a- b|| and ||a+ b||-||a- b|| Homework Equations...
  29. N

    All real numbers are complex numbers?And are I #'s orthogonal R#'s?

    A) I understand that complex numbers come in the form z= a+ib where a and b are real numbers. In the special case that b = 0 you get pure real numbers which are a subset of complex numbers. I read that both real and imaginary numbers are complex numbers so I am a little confused with notations...
  30. A

    Is sin(nπ) always equal to zero for integer n?

    Homework Statement Homework Equations The Attempt at a Solution I'm not sure how to prove that it is zero. I don't see what I can do after the second last step.
  31. A

    Proving a set of functions is orthogonal

    Why is the math in the red box necessary? According to this definition, it isn't:
  32. A

    What does it mean if two functions are orthogonal?

    I already know the definition: http://en.wikipedia.org/wiki/Orthogonal_functions But what does it mean intuitively, analytically, or in terms of graphs?
  33. K

    MHB Geometric action of an arbitrary orthogonal 3x3 matrix with determinant -1

    Hi, I have a question about describing geometrically the action of an arbitrary orthogonal 3x3 matrix with determinant -1. I would like to know if my proposed solutions are satisfactory, or if they lack justification. I have two alternate solutions, but have little confidence in their validity...
  34. D

    Proving that the orthogonal subspace is invariant

    Hi guys, I couldn't fit it all into the title, so here's what I'm trying to do. Basically, I have a unitary representation V. There is a subspace of this, W, which is invariant if I act on it with any map D(g). How do I prove that the orthogonal subspace W^{\bot} is also an invariant subspace...
  35. F

    A Definition of hypersurface orthogonal

    Definition of "hypersurface orthogonal" Hi all! I'm not sure if the thread belongs more to General Relativity or Differential Geometry, but I guess the border is labile. I've come across the term "hypersurface orthogonal" many times, but I still haven't found a clear definition. Apparently...
  36. K

    Proving E and B are orthogonal

    Homework Statement E . B =0 Homework Equations ∇xE=B The Attempt at a Solution I know AxB=C implies both A and B are orthogonal to C but does the same thing ring true for the Del cross something? In any case, is there a nice simple proof for the problem stated? This is not HW...
  37. A

    Expressing A for Orthogonal Vectors: Conditions on a,b & c

    Let a,b,c be three 3x1 vectors. Let A be a 3x3 upper triangular matrix which ensures that the 3x1 vectors d,e and f obtained using [d e f]=A[a b c] are orthogonal. a)Express the elements of A in terms of vectors a,b and c. b)what is the condition on a,b and c which allows us to find an...
  38. A

    If f and g are orthogonal, are f* and g orthogonal?

    I am curious: if f and g are (complex) orthogonal functions, are f* and g also orthogonal? (* denotes complex conjugate). I would think the answer is no, in general, but I just want to confirm
  39. C

    Orthogonal group/linear algebra/group theory

    Homework Statement This problem has two parts: i) Determine the range of det: O(n) → ℝ. ii) Are det-1({1})⊂O(n) and det-1({1})⊂O(n) groups? Homework Equations AA-1=I & AAT=I The Attempt at a Solution i) det(AAT)=det(I) det(AAT)=1 det(A) det(AT)=1 det(A) det(A)=1...
  40. S

    Proof regarding orthogonal projections onto spans

    Homework Statement Let U be the span of k vectors, {u1, ... ,uk} and Pu be the orthogonal projection onto U. Let V be the span of l vectors, {v1, ... vl} and Pv be the orthogonal projection onto V. Let X be the span of {u1, ..., uk, v1, ... vl} and Px be the orthogonal projection onto X...
  41. C

    Equivalent statements to matrix A is orthogonal

    A is orthogonal if the A^{-1} = A^{T}. Thus, AA^{T} = I. However, is the statement A is orthogonal equivalent to A^{2}=I. I don't think the statements are equivalent, but it comes from a test. Thus, I'd hope the test is correct.
  42. C

    What happens to the form basis after making the metric time orthogonal

    Given a basis for spacetime ##\{e_0, \vec{e}_i\}## for which ##\vec{e}_0## is a timelike vector. Of these vectors one can make a new basis for which all vectors are orthogonal to ##\vec{e}_0##. I.e. the vectors $$\hat{\vec{e}}_i = \vec{e}_i - \frac{\vec{e}_i \cdot \vec{e}_0}{\vec{e}_0 \cdot...
  43. evinda

    MHB Construct Orthogonal Basis in R^3: Solve Exercise

    Hello! I am stuck at the following exercise: "Construct an orthogonal basis of R^{3} (in terms of Euclidean inner product) that contains the vector \begin{pmatrix}2\\1 \\-1 \end{pmatrix} " What I've done so far is: Let {(a,b,c), (k,l,m), (2,1,-1)} be the basis. Then since the basis has to...
  44. Jameson

    MHB What are the properties of orthogonal matrices?

    Problem: Let $O$ be an $n \times n$ orthogonal real matrix, i.e. $O^TO=I_n$. Prove that: a) Any entry in $O$ is between -1 and 1. b) If $\lambda$ is an eigenvalue of $O$ then $|\lambda|=1$ c) $\text{det O}=1 \text{ or }-1$ Solution: I want to preface this with that although this is a 3-part...
  45. P

    Orthogonal eigenvectors and Green functions

    Hi you all. I have to diagonalize a hermitian operator (hamiltonian), that has both discrete and continuous spectrum. If ψ is an eigenvector with eigenvalue in the continuous spectrum, and χ is an eigenvector with eigenvalue in the discrete spectrum, is correct to say that ψ and χ are always...
  46. lonewolf219

    Significance of orthogonal polynomials

    Two polynomials are considered orthogonal if the integral of their inner product over a defined interval is equal to zero... is that a correct and complete definition? From what I understand, orthogonal polynomials form a basis in a vector space. Is that the desirable quality of orthogonal...
  47. lonewolf219

    Applications of orthogonal polynomials

    I was wondering if anyone could provide some examples of when/where the following orthogonal polynomials are used in physics? I'm starting a research project in the math department, and my professor is trying to steer the project back to physics, asking for specific applications of the...
  48. P

    Orthogonal projection - embarrassed

    Hi there I am trying to project some 3D points on to the span of two orthogonal vectors. v1 = [ -0.1235 -0.9831 0.1352] v2 = [ 0.7332 -0.1822 -0.6552] I used the orthogonal projection formula newpoint = oldpoint-dot(oldpoint,normal(v1,v2))*normal(v1,v2); but when I plot...
  49. E

    Difference between orthogonal transformation and linear transformation

    What is the difference between orthogonal transformation and linear transformation?
  50. Petrus

    MHB Show that diagonals in a diamond (rhombus) are orthogonal

    Hello MHB, I am working with a exemple that I think they got some incorrect. Exemple 10.Show that diagonals in a diamond(romb) is orthogonal. I understand that AC•BD=0 cause of dot product and if it's orthogonal the angle is \frac{\pi}{2} I understand all the part until the step before the last...
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