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. I Pullback & orthogonal projector

We have a map ##\phi : M \rightarrow N##, where ##N## has dimension ##n## and ##M## has dimension ##m=n-1##. So we consider the hypersurface ##\Sigma \equiv \phi(M)## picked out by the map. We also have an orthogonal projector, ##{\bot^a}_b \equiv \delta^a_b + n^a n_b##, where ##n## is the unit...
2. I Citation needed: Only multivariate rotationally invariant distribution with iid components is a multivariate normal distribution

I need a citation for the following proposition: Assume a random vector ##X=(X_1, ..., X_n)^T## with iid components ##X_i## and mean 0, then the distribution of ##X## is only invariant with respect to orthogonal transformations, if the distribution of the ##X_i## is a normal distribution. Thank...
3. I Why do orthogonal polarizers at slits eliminate interference pattern?

1) Very simple setup: a light source sends single photons towards a double-slit setup. After slit A there is a horizontal polarizer, and after slit B there is a vertical polarizer. Finally, there is a back screen. In this setup we will see no interference pattern, despite the fact that there...
4. B Magnetic field produced by an electric current

Hi everyone . if an alternating electric current passes through a piece of straight conducting wire, a proportional magnetic field appears on the orthogonal plane. what happens to the magnetic field if instead of copper, as a conductor, I use different materials with particular characteristics...
5. 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...
6. Square of orthogonal matrix vanishes

I found a the answer in a script from a couple years ago. It says the kinetic energy is $$T = \frac{1}{2} m (\dot{\vec{x}}^\prime)^2 = \frac{1}{2} m \left[ \dot{\vec{x}} + \vec{\omega} \times (\vec{a} + \vec{x}) \right]^2$$ However, it doesn't show the rotation matrix ##R##. This would imply...
7. B Question about orthogonal vectors and the cosine

Hi, The orthogonality defect is ##\prod_i ||b_i|| / det(B)##. Now it is said: The relation between this quantity and almost orthogonal bases is easily explained. Let ##\theta_i## be the angle between ##b_i## and ##span(b_1,...,b_{i-1})##. Then ##||b_i^*|| = ||b_i|| cos(\theta_i)##. [...] So...
8. I What's an example of orthogonal functions? Do these qualify?

Wiki defines orthogonal functions here https://en.wikipedia.org/wiki/Orthogonal_functions Here's one example, but it's an example that is only true for a specific interval https://www.wolframalpha.com/input?i=integral+sin(x)cos(x)+from+0+to+pi So are these functions orthogonal because there...
9. MHB Invariance of Asymmetry under Orthogonal Transformation

Show that the property of asymmetry is invariant under orthogonal similarity transformation
10. 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...
11. Are Orthogonal Vectors Proven by Derivative and Dot Product?

I feel like this question is very straight forward and my explanation below summarizes the answer pretty well. Could someone confirm this or tell me if I am missing something? We have V which is a vector, but the question states it is a constant. If I take the derivative of V, represented by...
12. Finding Orthogonal Matrices: 2 Solutions and Help

I have found two such matrices: ##\begin{pmatrix} -cos( \frac {\pi} {4}) & sin(\frac {\pi} {4})\\ sin(\frac {\pi} {4}) & cos(\frac {\pi} {4})\end{pmatrix}####\begin{pmatrix} -cos( \frac {\pi} {4}) & -sin(\frac {\pi} {4})\\ -sin(\frac {\pi} {4}) & cos(\frac {\pi} {4})\end{pmatrix}## Any hint...
13. Dimension of orthogonal subspaces sum

##| V_1 \rangle \in \mathbb{V}^{n_1}_1## and there is an orthonormal basis in ##\mathbb{V}^{n_1}_1##: ##|u_1\rangle, |u_2\rangle ... |u_{n_1}\rangle## ##| V_2 \rangle \in \mathbb{V}^{n_2}_2## and there is an orthonormal basis in ##\mathbb{V}^{n_2}_2##: ##|w_1\rangle, |w_2\rangle ...
14. Orthogonal Projection Problems?

Summary:: Hello all, I am hoping for guidance on these linear algebra problems. For the first one, I'm having issues starting...does the orthogonality principle apply here? For the second one, is the intent to find v such that v(transpose)u = 0? So, could v = [3, 1, 0](transpose) work?
15. MHB Set of 2-dimensional orthogonal matrices equal to an union of sets

Hey! :giggle: The set of $2$-dimensional orthogonal matrices is given by $$O(2, \mathbb{R})=\{a\in \mathbb{R}^{2\times 2}\mid a^ta=u_2\}$$ Show the following: (a) $O(2, \mathbb{R})=D\cup S$ and $D\cap S=\emptyset$. It holds that $D=\{d_{\alpha}\mid \alpha\in \mathbb{R}\}$ and...
16. B Orthogonal Projections: Same Thing or Not?

Aren't they the same thing? If so, why would textbooks write the former? Ex: https://textbooks.math.gatech.edu/ila/projections.html or http://www.math.lsa.umich.edu/~speyer/417/OrthoProj.pdf or https://en.wikipedia.org/wiki/Projection_(linear_algebra)#Orthogonal_projections Thank you!
17. How do I obtain a set of orthogonal polynomials up to the 7th term?

Hello everyone, I need some help with this solution. I'm trying to obtain a set of orthogonal polynomials up to the 7th term. I think i got it up to the 6th term, but the integration is getting more complex. I'm not sure if I'm on the right track. Please help
18. A Understanding the Relationship between Orthogonal and Unitary Groups

I'm a little bit confused. Matrices \begin{bmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{bmatrix} ##\theta \in [0,2\pi]## form a group. This is special orthogonal group ##SO(2)##. However it is possible to diagonalize this matrices and get \begin{bmatrix} e^{i\theta} & 0...
19. A Matrix multiplication, Orthogonal matrix, Independent parameters

Matrix multiplication is defined by \sum_{k}a_{ik}b_{kj} where ##a_{ik}## and ##b_{kj}## are entries of the matrices ##A## and ##B##. In definition of orthogonal matrix I saw \sum_{k=1}^n a_{ki}a_{kj}=\delta_{ij} This is because ##A^TA=I##. How to know how many independent parameters we have in...
20. MHB Proving Orthogonal Projection of Triangle V, v'_{1}

Given the triangle above where V < v'_{1}, prove that the $v_{1}=V \cos(\psi)+v'_{1} \cos(\theta - \psi)$ It is said that v_{1} is equal to the sum of the orthogonal projections on v_{1} of V and of v'_{1} and that is precisely the expression that show. But I couldn't see how to make the...
21. Using Least Squares to find Orthogonal Projection

I'm a little confused how to do this homework problem, I can't seem to obtain the correct answer. I took my vectors v1, v2, and v3 and set up a matrix. So I made my matrix: V = [ (6,0,0,1)T, (0,1,-1,0)T, (1,1,0,-6)T ] and then I had u = [ (0,5,4,0) T ]. I then went to solve using least...
22. Legendre Polynomials as an Orthogonal Basis

If you had legendre polynomials defined in ##L^2([-1,1])##, with ##||Pn_2||^2=\frac{2}{2n+1}##, show that for any polynomial with p a set of ##L^2([-1,1])##, with degree less than n, we have the inner product of ##P_n## and p = 0. Find the polynomials ##P_0,... P_4## Tried to use the integral...
23. The general equation of the superposition of orthogonal waves?

hi guys i was trying to derive the general formula of two orthogonal waves $$x^{2}-2xycos(Ξ΄)+y^{2} = A^{2} sin(Ξ΄)^{2}$$ where the two waves are given by : $$x = Acos(Οt)$$ $$y = Acos(Οt+Ξ΄)$$ where ##Ξ΄## is the different in phase , i know it seems trivial but i am stuck on where should i begin...
24. I Riemannian Fisher-Rao metric and orthogonal parameter space

Let ## \mathcal{S} ## be a family of probability distributions ## \mathcal{P} ## of random variable ## \beta ## which is smoothly parametrized by a finite number of real parameters, i.e., ## \mathcal{S}=\left\{\mathcal{P}_{\theta}=w(\beta;\theta);\theta \in \mathbb{R}^{n}...
25. Orthogonal trajectories in polar coordinates

there is a problem in a book that asks to find the orthogonal trajectories to the curves described by the equation : $$r^{2} = a^{2}\cos(\theta)$$ the attempt of a solution is as following : 1- i defferntiate with respect to ##\theta## : $$2r \frac{dr}{d\theta} = -a^{2}\;\sin(\theta)$$ 2- i...
26. A Inhomogeneous wave equation: RHS orthogonal to homogeneous solutions

Hi, I've been reading Brillouin's 'Wave Propagation in Periodic Media'. About the following equation $$\nabla^2u_1+\frac{\omega^2_0}{V_0}u_1 = R(r)$$ Brillouin states that "it is well known that such an equation possesses a finite solution only if the right-hand term is orthogonal to all...
27. B The nature of orthogonal oscillations (extending E&M)

Classical electromagnetic propagation evokes an electric field at right angles to a magnetic field. Does this complementary directionality have a simpler basis in QED? Are there any examples of an orthogonal component in other fundamental interactions? Thanks.
28. Minimum time between two orthogonal states

E = (1/β2)^2(E1) + (1/β2)^2(E2) = (E1+E2)/2 Let Ο(x,t=0) = Ο0 So, Ο1 = Ο0*exp(-i*E*T1/Δ§) and, Ο2 = Ο0*exp(-i*E*T2/Δ§) Given, <Ο1|Ο0> = <Ο2|Ο0> = 0 So, <Ο0*exp(-i*E*T1/Δ§)|Ο0> = 0 => exp(i*E*T1/Δ§)<Ο0|Ο0> = 0 => exp(i*E*T1/Δ§) = 0 Similarly, exp(i*E*T2/Δ§) = 0 So, exp(i*E*T1/Δ§) = exp(i*E*T2/Δ§)...
29. B Why is "time orthogonal to space" in inertial reference frames?

I'm reading about the geometry of spacetime in special relativity (ref. Core Principles of Special and General Relativity by Luscombe). Here's the relevant section: ----- Minkowski space is a four-dimensional vector space (with points in one-to-one correspondence with those of ##\mathbb{R}^4##)...
30. I Proof that two timelike vectors cannot be orthogonal

For fun, I decided to prove that two timelike never can be orthogonal. And for this, I used the Cauchy Inequality for that. Such that The timelike vectors defined as, $$g(\vec{v_1}, \vec{v_1}) = \vec{v_1} \cdot \vec{v_1} <0$$ $$g(\vec{v_2}, \vec{v_2}) = \vec{v_2} \cdot \vec{v_2} <0$$ And the...
31. Subspace of vectors orthogonal to an arbitrary vector.

The proof that the set is a subspace is easy. What I don't get about this exercise is the dimension of the subspace. Why is the dimension of the subspace ##n-1##? I really don't have a clue on how to go through this.
32. E

Eigenvalues of an orthogonal matrix

I'm fairly stuck, I can't figure out how to start. I called the matrix ##\mathbf{A}## so then it gives us that ##\mathbf{A}\mathbf{A}^\intercal = \mathbf{I}## from the orthogonal bit. I tried 'determining' both sides... $$(\det(\mathbf{A}))^{2} = 1 \implies \det{\mathbf{A}} = \pm 1$$I don't...
33. Resultant of two orthogonal vectors

But the answer in my book is given that sec(theta) =3. Where am I going wrong?
34. B How do orthogonal waves interfere?

How do two highly directional, orthogonal light beams (or any other kind of waves) with the same frequency interfere with each other?
35. A Orthogonal complement of the orthogonal complement

Consider the infinite dimensional vector space of functions ##M## over ##\mathbb{C}##. The inner product defined as in square integrable functions we use in quantum mechanics. If we already know that the orthogonal complement is itself closed, how can we show that the orthogonal complement of...
36. A Orthogonal spacelike and timelike vectors and inertial frames

I know that any vector ##V## in Minkowski spacetime can be classified in three different categories based on its norm ##|V| = \sqrt{V \cdot V} = V^{\mu}V_{\mu}##. These are: 1) If ##V^{\mu}V_{\mu} < 0##, ##V^{\mu}## is timelike. 2) If ##V^{\mu}V_{\mu} > 0##, ##V^{\mu}## is spacelike. 3) If...
37. E

Why doesn't using a basis which is not orthogonal work?

As far as I know, a set of vectors forms a basis so long as a linear combination of them can span the entire space. In ##\mathbb{R}^{2}##, for instance, it's common to use an orthogonal basis of the ##\hat{x}## and ##\hat{y}## unit vectors. However, suppose I were to set up a basis (again in...
38. 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...
39. Covering of the orthogonal group

Progress:π:π(3)ββ€2π:π(3)βππ(3)π:π(3)/ππ(3)ββ€2 π(π)=det(π) with πβπ(3), that way π(π)β¦{β1,1}ββ€2, where 1 is the identity element.Ker(π) = {πβππ(3)|π(π)=1}=ππ(3), since det(π)=1 for πβππ(3).By the multiplicative property of the determinant function, π = homomorphism. ***What is the form of the...
40. I Proof of ##F## is an orthogonal projection if and only if symmetric

The given definition of a linear transformation ##F## being symmetric on an inner product space ##V## is ##\langle F(\textbf{u}), \textbf{v} \rangle = \langle \textbf{u}, F(\textbf{v}) \rangle## where ##\textbf{u},\textbf{v}\in V##. In the attached image, second equation, how is the...
41. I Orthogonal state with m = 0 carries s = 0 .... explanation?

Hello I could use some help understanding a statement / sentence within my Griffiths Quantum Mechanics book. The same statement is made within video lecture I found surfing to understand the Griffiths text. I have the 2nd edition. (On page 185) Discussing addition of angular momenta β 2 spin Β½...
42. MHB Orthogonal Projections .... Garling, Proposition 11.4.3 .... ....

I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume II: Metric and Topological Spaces, Functions of a Vector Variable" ... ... I am focused on Chapter 11: Metric Spaces and Normed Spaces ... ... I need some help to fully understand the proof of...
43. B Why is time orthogonal to space?

Basically how do we know that since it is not possible to see the 4th dimension, is it for simplicity ?
44. MHB What are the orthogonal trajectories of e^{x}(xcosy - ysiny) = c?

Asalamoalaikum, help me with this. I can solve it but it goes very lengthy. Determine the equations of the orthogonal trajectories of the following family of curve; e^{x}(xcosy - ysiny) = c
45. M

I Proving the Orthogonal Projection Formula for Vector Subspaces

Hi PF! I've been reading and it appears that the orthogonal projection of a vector ##v## to the subspace spanned by ##e_1,...,e_n## is given by $$\sum_j\langle e_j,v \rangle e_j$$ (##e_j## are unit vectors, so ignore the usual inner product denominator for simplicity) but there is never a proof...
46. I Orthogonal transformation and mirror transformation

How to prove any orthogonal transformation can be represented by the product of many mirror transformations, please?What's the intuitive meaning of this proposition? Thank you.
47. MHB Orthogonal vector projection and Components in Orthogonal Directions ....

I am reading Miroslav Lovric's book: Vector Calculus ... and am currently focused n Section 1.3: The Dot Product ... I need help with an apparently simple matter involving Theorem 1.6 and the section on the orthogonal vector projection and the scalar projection ...My question is as follows: It...
48. Finding Orthogonal Trajectories (differential equations)

Homework Statement Find Orthogonal Trajectories of ##\frac{x^2}{a}-\frac{y^2}{a-1}=1## Hint Substitute a new independent variable w ##x^2=w## and an new dependent variable z ##y^2=z## Homework EquationsThe Attempt at a Solution substituting ##x## and ##y## I get...
49. I Understanding what the complex cosine spectrum is showing

The complex exponential form of cosine cos(k omega t) = 1/2 * e^(i k omega t) + 1/2 * e^(-i k omega t) The trigonometric spectrum of cos(k omega t) is single amplitude of the cosine function at a single frequency of k on the real axis which is using the basis function of cosine, right? The...
50. Orthogonal projection onto a plane spanned by two vectors

Homework Statement x = <0, 10, 0> v1 = <4, 3, 0> v2 = <0, 0, 1> Project x onto plane spanned by v1 and v2 Homework Equations Projection equation The Attempt at a Solution I took the cross product k = v1xv2 = <3, -4, 0> I projected x onto v1xv2 [(x*k)/(k*k)]*k = <-4.8, 6.4, 0 = p I finished...