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. J

    Orthogonality of eigenfunctions with continuous eigenvalues

    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...
  2. F

    Orthogonality Proof with Differentiable Vector Function

    Homework Statement Suppose \Pi \subset \mathbb{R}^3 is a plane, and that P is a point not on \Pi . Assume that Q \in \Pi is a point on \Pi whose distance to P is minimal. Show that the vector PQ is orthogonal to \Pi . Define a differentiable vector function r(t) with r(t) \in \Pi and...
  3. S

    Orthogonality: intuition challenged.

    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...
  4. M

    Linear Algebra: Orthogonality of Hermitean Projectors

    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...
  5. W

    Orthogonality of Two Functions

    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...
  6. nomadreid

    Maxwell's equations, Orthogonality, electric and magnetic fields in EM

    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?
  7. A

    Proving Orthogonality of Two Families of Parabolas

    Please,help me with this problem. Homework Statement Prove that the two families of parabolas y^2=4a(a-x),a>0 and y^2=4b(b+x),b>0 form an orthogonal net. Specifically, check that for any a, b > 0 these two parabolas are perpendicular to each other at the points where they intersect. The...
  8. E

    Orthogonality of momentum space wavefunctions

    Page 152 Robinett: Consider the (non-normalized) even momentum space wavefunctions for the symmetric well:, \phi_n^+(p) = 2sin(w-m)/(w-m)+sin(w+m)/(w+m) where w = sin((n-1/2)pi) and m = ap/hbar. Show that \int_{-\infty}^{\infty}\phi_n^+(p)^*\cdot \phi_n^+(p) dp = \delta_{n,m} The hint...
  9. C

    Are all Killing Vectors in 2D Spacetime Hypersurface Orthogonal?

    Just a quick question. Suppose we are considering a (1+1) spacetime. Are all vector fields hypersurface orthogonal? I think the answer is yes, since in the formula \xi_{[a}\partial_b \xi_{c]}, two indices will always be the same, which I think automatically makes the expression equal to zero...
  10. R

    Orthogonality in Basis Sets: Exploring the Overlap of Atomic Orbitals

    1: Why are the elements of a basis set taken to be orthogonal? But in real sense atomic orbitals do overlap.
  11. N

    Eigenfunctions (orthogonality & expansion)

    1) If you have a particle in 1D bound within range "-a" and "a". You come up with one eigenfunction that is sinusoidal (since it satisfies the problem). Now, you get all the necessary constants through the usual way... I want to know whether more than one eigenfunction can be produced and...
  12. K

    Orthogonality of 1s and 2s Orbitals of H

    Homework Statement show that 1s and 2s orbitals of H are orthogonal Homework Equations orbital functions n=1 and n=2 The Attempt at a Solution Im asking what values(range) should i integrate the two equations into. thank you
  13. L

    Orthogonality relations of functions e^(2 pi i n x)

    I know that the functions e^{2 \pi inx} for n \in \mathbb{Z} are a base in the space of functions whith period 1. How do I derive the orthogonality relations for these functions?
  14. L

    Proving Orthogonality of Legendre Polynomials

    Problem: Show that \int_{-1}^{1} x P_n(x) P_m(x) dx = \frac{2(n+1)}{(2n+1)(2n+3)}\delta_{m,n+1} + \frac{2n}{(2n+1)(2n-1)}\delta_{m,n-1} I guess I should use orthogonality with the Legendre polynomials, but if I integrate by parts to get rid of the x my integral equals zero. Any tip on...
  15. S

    Orthogonality Proof for R(x)*Wn*Yn(x) = -T(d^2)[Yn(x)]/dx

    We have the eigenvalue equation R(x)*(Wn*2)*Yn(x) = -T (d^2)[Yn(x)]/dx T the tension is a constant R(x) is the weight function, Wn is the eigenvalue of the nth normal mode and Yn is the nth normal mode eigenfunction. We have to show that the integral of (Yn(x)*Ym(x)*R(x)) between the...
  16. I

    Issue regarding the orthogonality of eigenvectors for Hermitian

    At the risk of arrousing the ire of the moderaters for posting the same topic in two forums, I again ask this question as no one in the quantum forum seems to be able to help. So... Regarding a proof of the orthogonality of eigenvectors corresponding to distinct eigenvalues of some Hermitian...
  17. I

    Problem with proof of orthogonality of eigenvectors for Hermitian

    I'm not sure if this is the appropriate section, perhaps my question is better suited for Linear Algebra. At any rate, here goes. Regarding a proof of the orthogonality of eigenvectors corresponding to distinct eigenvalues of some Hermitian operator A: Given A|\phi_1\rangle = a_1|\phi_1\rangle...
  18. F

    Clarifying Orthogonal Vectors: Understanding Homework Notation

    I'm confused on the following questions. (1) Find a vector that is perpendicular to (v_1,v_2). (2) Find two vectors that are perpendicular to (v_1,v_2,v_3. This homework set was written by the professor (it is review) before we actually get into the new material. The notation the book uses is...
  19. M

    Proof of orthogonality of associated Legendre polynomial

    Hi, I'm trying to prove the orthogonality of associated Legendre polynomial which is called to "be easily proved": Let P_l^m(x) = (-1)^m(1-x^2)^{m/2} \frac{d^m} {dx^m} P_l(x) = \frac{(-1)^m}{2^l l!} (1-x^2)^{m/2} \frac {d^{l+m}} {dx^{l+m}} (x^2-1)^l And prove \int_{-1}^1...
  20. T

    3 by 3 matrix with an orthogonality constraint

    This is a paragraph from a book, which I don't understand: "How many independent parameters are there in a 3x3 matrix? A real 3x3 matrix has 9 entries but if we have the orthogonality constraint, RR^T = 1 which corresponds to 6 independent equations because the product RR^T being the same...
  21. mattmns

    Proving Orthogonality of Non-Zero Vectors

    For non-zero vectors v and w show that ||v||w + ||w||v is orthogonal to ||v||w - ||w||v I am baffled by this problem, I know that a way to solve it would be to say that the first dot the second = 0, but I am just unable to prove it for all cases from there. I know it is true though. Any...
  22. E

    Exploring Orthogonality in Linear Algebra: A Simple Proof of a Common Identity

    In one of my homework problems we are asked to prove this: |X + Y|^2 + |X - Y|^2 = 2|X|^2 + 2|Y|^2 It seems quite simple: I just expanded both of them (cancelled 2(X dot Y) terms) and came up with the expression on the right. Is there something I am missing or is it really that simple...
  23. H

    Proving Orthogonality of Product of Matrices

    How do you prove that the product of two orthogonal matrices is orthogonal? I know that a matrix can be written in component form as A=a_{jk} and that for an orthogonal matrix, the inverse equals the transpose so a_{kj}=(a^{-1})_{jk} and matrix multiplication can be expressed as...
  24. T

    Are Even and Odd Functions Orthogonal?

    We were doing examples in class today and showed that sin and cos were orthogonal functions. In general, is true that even and odd functions are orthogonal? I was unsure where a proof of this might begin, mostly how to generalize the notion of an even or odd function.
  25. B

    Quantum Mechanics: Eigenvaules, and orthogonality

    Hey, I've been trying to solve this problem it sounds simple but i don't know where to start: If \phi_{1} and \phi_{2} are normalised, have the same eigenvalue and obey \int \phi_{1}*\phi_{2}d\tau = c find the linear combination that is normalised and orthogonal to \phi_{1} Thanks
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