# 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. ### Deduce orthogonality relations for sine and cosine w/ Euler's Formula

To solve part (a), we write ##e^{inx}e^{-imx}=e^{ix(n-m)}##. If ##m=n## then this expression is 1, and so the integral of 1 from 0 to ##2\pi## is ##2\pi##. If ##m\neq n## then we use Euler's formula and integrate. The result is zero. My question is how do we solve part (b) using part (a)? I...
2. ### Orthogonality of 2 complex exponentials

For 2 complex functions, to find the orthogonality, one of the function has to be in complex conjugate? Because in the lecture note, the first formula is without complex conjugate, so I’m a bit confused
3. ### Prove orthogonality of these curves

I am asked to prove orthogonality of these curves, however my attempts are wrong and there's something I fundamentally misunderstand as I am unable to properly find the graphs (I have only found for a, but I doubt the validity). Furthermore, I am familiar that to check for othogonality (based...
4. ### I About global inertial frame in GR - revisited

Hi, reading this old thread I'd like a clarification about the following: Fermi Normal hypersurface at an event on a comoving FLRW worldline is defined by the collection of spacetime orthogonal geodesics. Such geodesics should be spacelike since they are orthogonal to the timelike comoving...
5. ### B Orthogonality in Minkowski Spacetime: Meaning & Visualization

I have read that non-inertial frames are those, where time is not orthogonal on space. Does it just mean that the speed of light is not isotropic there or does it mean anything else? How can I picture more easily this concept (for space orthogonality I just imagine perpendicularity of one axis...
6. ### I Finding the orthogonal projection of a vector without an orthogonal basis

Hi there, I am currently reading a course on euclidian spaces and I came across this result that I am struggling to prove : Let ##F## be a subspace of ##E## (of finite dimension) such that ##F=span(e_1, e_2, ..., e_p)## (not necessarily an orthogonal family of vectors), let ##x \in E## Then...
7. ### I Orthogonality of Eigenvectors of Linear Operator and its Adjoint

Suppose we have V, a finite-dimensional complex vector space with a Hermitian inner product. Let T: V to V be an arbitrary linear operator, and T^* be its adjoint. I wish to prove that T is diagonalizable iff for every eigenvector v of T, there is an eigenvector u of T^* such that <u, v> is...
8. ### A Orthogonality of variations in Faddev-Popov method for path integral

Hi there, I've been stuck on this issue for two days. I'm hoping someone knowledgeable can explain. I'm working through the construction of the quantum path integral for the free electrodynamic theory. I've been following a text by Fujikawa ("Path Integrals and Quantum Anomalies") and also...
9. ### I Similarity transformation, basis change and orthogonality

I've a transformation ##T## represented by an orthogonal matrix ##A## , so ##A^TA=I##. This transformation leaves norm unchanged. I do a basis change using a matrix ##B## which isn't orthogonal , then the form of the transformation changes to ##B^{-1}AB## in the new basis( A similarity...
10. ### I Synchronous Reference Frame: Definition and Usage

Hi, reading the Landau book 'The Classical theory of Field - vol 2' a doubt arised to me about the definition of synchronous reference system (a.k.a. synchronous coordinate chart). Consider a generic spacetime endowed with a metric ##g_{ab}## and take the (unique) covariant derivative operator...
11. ### I Discrete Orthogonality Relations for Cosines

Hi all, I've come across some problem where I have terms such as ##\sum_{j=1}^N \cos(2 \pi j k /N) \cos(2 \pi j k' /N)##, or ##\sum_{j=1}^N \cos(2\pi j k/ N)##, or ## \sum_{j=1}^N \cos(2\pi j k/ N) \cos(\pi j) ##. In all cases we have the extra condition that ##1 \le k,k' \le N/2-1## (and...
12. ### Orthogonality Relationship for Legendre Polynomials

Suppose p = a + bx + cx². I am trying to orthogonalize the basis {1,x,x²} I finished finding {1,x,x²-(1/3)}, but this seems different from the second legendre polynomial. What is the problem here? I thought could be the a problem about orthonormalization, but check and is not.
13. ### On the orthogonality of the rotation matrix

Let me start with the rotated vector components : ##x'_i = R_{ij} x_j##. The length of the rotated vector squared : ##x'_i x'_i = R_{ij} x_j R_{ik} x_k##. For this (squared) length to be invariant, we must have ##R_{ij} x_j R_{ik} x_k = R_{ij} R_{ik} x_j x_k = x_l x_l##. If the rotation matrix...
14. ### I Orthogonal eigenvectors and measurement

An outcome of a measurement in a (infinite) Hilbert space is orthogonal to all possible outcomes except itself! This sounds related to the measurement problem to me, for we inherently only obtain a single outcome. So, to take a shortcut I posted this question so I quickly get to hear where I'm...
15. ### I Homogeneous equation and orthogonality

Hi, I'm going to cite a book that I'am reading Can anyone provide some simple references where I can find at least an intuition regarding what is stated by the author. Thanks, Ric
16. ### I QM Orthogonality: Separate & Independent Eigenvalues?

In non-relativistic QM, given a Hilbert Space with a Hermitian operator A and a generic wave function Ψ. The operator A has an orthogonal eigenbasis, {ai}. I have often read that the orthogonality of such eigenfunctions is an indication of the separateness or distinctiveness of the associated...
17. ### I Why does a quantum system with many degrees of freedom imply orthogonality?

Quantum decoherence means that when a quantum system interacts with its environment, coherence is lost, which means that all the density matrix becomes diagonal after the interaction. I never understood why it is so, but I get a clue here...
18. ### 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...
19. ### I Understanding Orthogonality: QM Lectures by Dr. Physics A

I need to understand orthogonality. I am monitoring QM lectures by Dr. Physics A, and he said all basis states of a state are orthogonal. I can understand that for the topics like polarization or spin, where Cartesian coordinates obtain with reference to measurements in one of 3...
20. ### Orthogonality of Gravitational Wave Polarizations

Homework Statement Two plane gravitational waves with TT (transverse-traceless) amplitudes, ##A^{\mu\nu}## and ##B^{\mu\nu}##, are said to have orthogonal polarizations if ##(A^{\mu\nu})^*B_{\mu\nu}=0##, where ##(A^{\mu\nu})^*## is the complex conjugate of ##A^{\mu\nu}##. Show that a 45 degree...
21. ### Verify orthogonality integral by direct integration

This is a heat equation related math problem. 1. Homework Statement The complete question is: Verify the orthogonality integral by direct integration. It will be necessary to use the equation that defines the λ_n: κ*λ_n*cos(λ_n*a) + h*sin(λ_n*a)=0. Homework Equations κ*λ_n*cos(λ_n*a) +...
22. ### I Orthogonality of Momentum Eigenstates: Why is ##\int e^{-ix}e^{2ix}dx=0##?

Consider two momentum eigenstates ##\phi_1## and ##\phi_2## representing momenta ##p_1## and ##p_2##. For the sake of easy numbers, ##p_1=1*\hbar## (with ##k=1##) and ##p_2=2*\hbar## (with ##k=2##). Thus, ##\phi_1=e^{ix}## and ##\phi_2=e^{2ix}##. Orthogonality states that ##\int...
23. ### I What is the relationship between dot products and orthogonality of functions?

first of all assume that I don't have proper math knowledge. I came across this idea while I was studying last night so I need to verify if it's valid, true, have sense etc. orthogonality of function is defined like this: https://en.wikipedia.org/wiki/Orthogonal_functions I wanted to...
24. ### Show orthogonality of vector-valued functions

I have this exercise on my book and I believe it is quite simple to solve, but I'm not sure if I did good, so here it is Homework Statement given a vector B ∈ ℝn, B ≠ 0 and a function F : ℝ → ℝn such that F(t) ⋅ B = t ∀t and the angle φ between F'(t) and B is constant with respect to t, show...
25. ### I Orthogonality of spherical Bessel functions

at what value of k should the following integral function peak when plotted against k? I_{\ell}(k,k_{i}) \propto k_{i}\int^{\infty}_{0}yj_{\ell}(k_{i}y)dy\int^{y}_{0}\frac{y-x}{x}j_{\ell}(kx)\frac{dx}{k^{2}} This doesn't look like any orthogonality relationship that I know, it's a 2D...
26. ### Linear combination and orthogonality

Given the non-zero vectors u, v and w in ℝ3 Show that there is a non-zero linear combination of u and v that is orthogonal to w. u and v must be linearly independant. I am not really sure at all. But I have done this: This is a screenshot of what I have done. Basicly, I assumed in the end...
27. ### A Bessel decomposition for arbitrary function

Orthogonality condition for the 1st-kind Bessel function J_m $$\int_0^R J_m(\alpha_{mp})J_m(\alpha_{mq})rdr=\delta_{pq}\frac{R^2}{2}J_{m \pm 1}^2(\alpha_{mn}),$$ where α_{mn} is the n^{th} positive root of J_m(r), suggests that an original function f(r) could be decomposed into a series of 1-st...
28. ### 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...
29. ### Orthogonality of Stationary States

Homework Statement I'm going back through some homework as revision, and came across this problem. It was marked as correct, but now I'm thinking it's unconvincing... For a particle in an infinite square well, with ##V = 0 , 0 \leq x \leq L##, prove that the stationary eigenstates are...
30. ### Trying to understand(conceptually) orthogonality

Homework Statement if Ax = b has a solution and A^Ty = 0 , is y^T(x) = 0 or y^T(y) =0 Homework EquationsThe Attempt at a Solution I simply do not think i understand the properties to answer this question. From my understandinging, the transpose of A times y is = 0. This means that A transpose...
31. ### Completeness and orthogonality of unitary irreducible representations

I'm reading Lie Algebras and Particle Physics by Howard Georgi. He is trying to prove (section 1.12) that the matrix elements of the unitary irreducible representations (irreps) form a vector space of dimension N where N is the order of the group. For example for the matrix of the kth unitary...
32. ### Modes orthogonality in a dielectric slab

A typical mode in a dielectric slab like this, with propagation along x, uniformity along z and refractive index variation along y, is represented by the following function: f (y) = \begin{cases} \displaystyle \frac{\cos (k_1 y)}{\cos (k_1 d)} && |y| \leq d \\ e^{-j k_2 (y - d)} && |y| \geq d...
33. ### Orthogonality of Wannier functions

I have trouble reconciling orthogonality condition for Wannier functions using both continuous and discrete k-space. I am using the definition of Wannier function and Bloch function as provided by Wikipedia (https://en.wikipedia.org/wiki/Wannier_function). Wannier function: Bloch function: I...
34. ### Orthogonality on Inner Product (Quantum Mechanics also)

Homework Statement Consider a qubit in the state |v> ∈ ℂ^2. Suppose that a measurement of δn is made on the qubit. Show that the probability of obtaining the result "+1" in the measurement is equal to 0 if and only if |v> and |n,+> are orthogonal. Homework Equations Inner product axioms |v>|w>...
35. ### Sturm-Liouville Orthogonality of Eigenfunctions

Homework Statement Consider the following Sturm-Liouville Problem: \dfrac{d^2y(x)}{dx^2} + {\lambda}y(x)=0, \ (a{\geq}x{\leq}b) with boundary conditions a_1y(a)+a_2y{\prime}(a)=0, \ b_1y(b)+b_2y{\prime}(b)=0 and distinguish three cases: a_1=b_1, a_2{\neq}0, b_2{\neq}0a_2=b_2=0, a_1{\neq}0...
36. ### Linear combination and orthogonality

Homework Statement Given the vectors a = (5,2,-1), b = (3,2,1), c = (1,2,3), b' = (1,1,0), c' = (3,-3,-2) We assume that the vector a is a linear combination of the vectors b and c and b' and c' respectively, so that: a = xb + yc = x'b' + y'c' a) Determine the factors x and y through...
37. ### Orthogonality and Orthonormality: Take 2

Homework Statement Which of the following sets of vectors in ℂ^3 is an orthogonal set? Which is an orthonormal set? Which is an orthonormal basis? \begin{pmatrix} 1/\sqrt{2}\\ 0\\ 1/\sqrt{2} \end{pmatrix}, \begin{pmatrix} -1/\sqrt{2}\\ 0\\ 1/\sqrt{2} \end{pmatrix}, \begin{pmatrix} 0\\...

40. ### Orthogonality of a curvilinear coordinate system

Homework Statement Show that the uvw-system is orthogonal. r, \theta, \varphi are spherical coordinates. $$u=r(1-\cos\theta)$$ $$v=r(1+\cos\theta)$$ $$w=\varphi$$ The Attempt at a Solution So basically I want to show that the scalar products between \frac{\partial \vec{r}}{\partial u}...
41. ### Orthogonality of inner product of generators

Hi, this is a rather mathematical question. The inner product between generators of a Lie algebra is commonly defined as \mathrm{Tr}[T^a T^b]=k \delta^{ab} . However, I don't understand why this trace is orthogonal, i.e. why the trace of a multiplication of two different generators is always zero.
42. ### Orthogonality from infinitesimal small rotation

Hello buddies, Could someone please help me to understand where the second and the third equalities came from? Thanks,
43. ### Orthogonality relations for Hankel functions

Where can I find and how can I derive the orthogonality relations for Hankel's functions defined as follows: H^{(1)}_{m}(z) \equiv J_{n}(z) +i Y_{n}(z) H^{(2)}_{m}(z) \equiv J_{n}(z) - i Y_{n}(z) Any help is greatly appreciated. Thanks
44. ### Orthogonality of Associated Laguerre Polynomial

I have a problem when trying to proof orthogonality of associated Laguerre polynomial. I substitute Rodrigue's form of associated Laguerre polynomial : to mutual orthogonality equation : and set, first for and second for . But after some step, I get trouble with this stuff : I've...
45. ### Fourier Series and orthogonality

Can someone explain the concept to me. Does it mean the the a's of n and b's of n are 90 degrees apart? I know the inner-product of the integral is 0 if the two are orthogonal.
46. ### On the orthogonality of Sturm-Liouville eigenvectors

From what I understand, solutions to the Sturm-Liouville differential equation (SLDE) are considered to be orthogonal because of the following statement: \left( \lambda_m-\lambda_n \right) \int_a^b w(x) y_m(x)y_n(x) dx = 0 My first question involves the assumptions that go into this...
47. ### On the orthogonality of Sturm-Liouville eigenvectors

From what I understand, solutions to the Sturm-Liouville differential equation (SLDE) are considered to be orthogonal because of the following statement: \left( \lambda_m-\lambda_n \right) \int_a^b w(x) y_m(x)y_n(x) dx = 0 My first question involves the assumptions that go into this...
48. ### Fourier series and orthogonality, completeness

http://ms.mcmaster.ca/courses/20102011/term4/math2zz3/Lecture1.pdfOn pg 10, the example says f(x)=/=0 while R.H.S is zero. It is an equations started from the assumption in pg 9; f(x)=c0f(x)0+c1f(x)1…, then how do we get inequality? if the system is complete and orthogonal, then...
49. ### How can two atomic orbitals with the same n value be orthogonal?

I am wondering how two orbitals of same n values can be orthogonal, for example how are a 2s and 3s orbital orthogonal? What I understand is a property of orthogonality is the product of the two wave functions integrate to zero over all space. I tried to look at this graphically and categorize...
50. ### Orthogonality of eigenstates and hermitian statement

if I derive a hermitian relation use: [1] \left \langle \Psi _{m} | H |\Psi _{n}\right \rangle =E_{n}\left \langle \Psi _{m} |\Psi _{n}\right \rangle and [2] \left \langle \Psi _{n} | H |\Psi _{m}\right \rangle =E_{m}\left \langle \Psi _{n} |\Psi _{m}\right \rangle if i take the complex...