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.
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
##| 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 ...
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?
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...
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!
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
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...
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...
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...
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...
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...
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 :
$$x = Acos(Οt)$$
$$y = Acos(Οt+Ξ΄)$$
where ##Ξ΄## is the different in phase , i know it seems trivial but i am stuck on where should i begin...
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}...
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...
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...
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.
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##)...
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...
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.
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...
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...
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...
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...
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...
Progress:π:π(3)ββ€2π:π(3)βππ(3)π:π(3)/ππ(3)ββ€2
π(π)=det(π)
with πβπ(3), that way
π(π)β¦{β1,1}β β€2,
where 1 is the identity element.Ker(π) = {πβππ(3)|π(π)=1}=ππ(3), since det(π)=1 for πβππ(3).By the multiplicative property of the determinant function, π = homomorphism.
***What is the form of the...
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...
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...
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...
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
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...
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.
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...
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...
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...
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...