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.
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...
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
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...
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...
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...
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...
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...
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...
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...
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...
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...
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.
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...
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...
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
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...
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...
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...
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...
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...
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) +...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...
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>...
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...
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\\...
Functions u, v satisfy the S-L eqtn $ [py']'+\lambda wy=0 $. u,v satisfy boundary conditions that lead to orthogonality. Prove that for appropriate boundary conditions, u' and v' are orthogonal with p as weighting factor.
I'm sure I need to use the orthogonality integral $ \langle u'|v'...
The book states that the integral of the (inner) product of 2 distinct eigenvectors must vanish if they are orthogonal.
Given $ P_1(x)=x, Q_0(x)= \frac{1}{2}\ln\left({\frac{1+x}{1-x}}\right) $ are solutions to Legendres eqtn., evaluate their orthog. integral.
Using $...
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}...
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.
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
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...
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.
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...
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...
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...
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...
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...