The Stack Overflow podcast is back! Listen to an interview with our new CEO.

Questions tagged [linear-algebra]

To be used for linear algebra, and closely related disciplines such as tensor algebras and maybe clifford algebras.

Filter by
Sorted by
Tagged with
3
votes
2answers
51 views

Differentiation of the determinant $g$

Let $g$ be the determinant of the metric tensor. I want to derive the following equation $g_{,\nu}=gg^{\lambda \mu}g_{\lambda \mu,\nu}$. It is said that $gg^{\lambda \mu}$ is a cofactor, but I can't ...
1
vote
3answers
152 views

A form $F$ is simple if and only if $F\wedge F=0$?

Gravitation by Charles W. Misner, Kip Throne and John Wheeler page 93 Box 4.1 point 5 b. Applications: a. "In four dimensions, all 0 forms, 1- forms, 3-forms, and 4-forms are simple. A 2-form $F$ is ...
0
votes
0answers
23 views

Hermitian phase operator and quantum harmonic oscillator

I need to apply a hermitian phase operator $\dfrac{1}{\sqrt{1+\hat{a}^\dagger\hat{a}}}\hat{a}$ to the nth harmonic oscillator state, but I have no idea how to interpret the square root of a ...
0
votes
0answers
33 views

How to find the hermitian adjoint and inverse of an operator?

Suppose I have a translation operator defined as: $$ \hat{T_a}\Psi(x)=\Psi(x+a) \, . $$ Now, how do I find the hermitian adjoint operator as well as its inverse?
0
votes
1answer
45 views

Photon near a black hole - find distance of closest approach from impact parameter

I have the equation relating the impact parameter $b$ to the distance of closest approach $R$. $R^3 - b^2R + 1 = 0$ which can be solved in python. I have a given $b$ and have to find $R$. however, ...
0
votes
0answers
9 views

Taking out % contribution of a zero order state from an eigenvector - dipole calculation

I am doing an analysis of a theoretical spectroscopy calculation. I take an eigenvector (nx1) and dot it with many zero-order dipole vectors (nx3) to get the dipole contribution to my new eigenstate. ...
0
votes
3answers
74 views

Can an eigenvalue be a function?

When we say that $$\hat{E}(\psi(x))=\alpha\psi(x),$$ where $\hat{E}$ is an operator and $\alpha$ is the eigenvalue. Is $\alpha$ a fixed constant(like a number) or can it's value keep on varying? ...
0
votes
0answers
16 views

Infinite Coupled Masses, symmetry, and the simultaneous diagonal theorem for infinite dimensional vector spaces

In The Physics of Waves by Georgi, in Chapter 4, we show that, in a coupled system of masses connected by springs, a transformation that preserves some symmetry $S$ commutes with $K^{-1}M$. From my ...
9
votes
4answers
2k views

How does a linear operator act on a bra?

I'm studying QM from Shankar. He introduces linear operators and says that an operator is an instruction for transforming one ket into another. But then a few lines below he says operators can also ...
11
votes
5answers
460 views

Complex conjugate and transpose “with respect to a basis”

In my quantum mechanics notes, my teacher described the complex conjugate and transpose of a linear operator X as "with respect to an orthogonal basis." What does it mean to take a transpose or ...
2
votes
1answer
27 views

$O(p,q)$ as transformations that conserve quadratic form

Let us try to define $O(p,q)$ in two different ways, which I want to show their equivalence. Define the symmetric bilinear quadratic form $[\cdot ,\cdot]$ which is given by $$[x,y]=\langle x,gy\...
0
votes
0answers
32 views

Proof of skewsymmetry of electromagntic function in Minkowski spacetime

I have been studying special relativity from the Gregory Naber's book: "The geometry of Minkowski spacetime" and I found a very strange proof. In Section 2.1, just before of equation 2.1.2. the book ...
0
votes
1answer
43 views

Reading energy Eigenvalues from a Hamiltonian matrix for 1D harmonic oscillator

After a perturbation $V(x)$ added to the system, a matrix element $H_{nn}$ calculated in unperturbed Eigenstates for one-dimensional harmonic oscillator is given as: $$\epsilon \hbar \omega_0\begin{...
1
vote
0answers
22 views

Finding an equivalent shape for a given mass and 3 mass moments of inertia

So I apologize if this is just impossible, but I was wondering if there was a way to find say, the dimensions of a box of a given density that would have the same mass and moments of inertia of ...
0
votes
1answer
31 views

Moment of Inertia Tensor Terminology

I've learned about the moment of inertia tensor as a matrix that can be used to compute angular momentum, moment of inertia, etc. for a system. But why is it often described as a tensor instead of a ...
4
votes
2answers
333 views

Why are physicists so interested in irreps if in their non-block-diagonal form they mix all components of a vector?

Consider a group $\{G,\circ\}$, with elements $e,g_1,g_2,...$, represented by the matrices $\{D(e), D(g_1), D(g_2)...\}$. If all the matrices can be brought to block diagonal forms by a similarity ...
2
votes
1answer
45 views

Spherical polar coordinates in a tetrad frame

I am looking at a paper which writes the spatial components of a vector $S_i$ in terms of spherical polar coordinates w.r.t the local tetrad frame as (Eq 33 in the linked paper), $$ S_1 = s \sin \...
0
votes
1answer
44 views

Understanding completeness relation and writing Hamiltonian in matrix form

A three level system hamiltonian I found where it is written as: $$\frac{H}{\hbar}= \Omega_1|e\rangle \langle g_1| + \Omega_1^* |g_1\rangle \langle e| + \Omega_2|e\rangle \langle g_2|+ \Omega_2^* |...
0
votes
1answer
34 views

Change of basis in a Euclidean space

I am trying to compute the change in the contravariant components of a vector when the basis is changed from Cartesian (standard basis) to spherical polars. I understand that a general vector $\...
1
vote
1answer
66 views

Identity for the inverse metric tensor using its determinant

I would like to prove this relation: $$g^{\mu\nu} = \frac{1}{3!} \frac{1}{g} \epsilon^{\mu\rho\sigma\kappa}\epsilon^{\nu\alpha\beta\gamma} g_{\rho\alpha} g_{\sigma\beta} g_{\kappa\gamma}, \tag{1}$$ ...
1
vote
1answer
41 views

What are the advantages of working in Pauli basis? [closed]

What are the advantages of working in Pauli basis $(\sigma_0, \sigma_1,\sigma_2, \sigma_3)$, in comparision to the natural basis? Here, $\sigma_0$ is the $2\times2$ identity matrix, and $\sigma_i$ $i=...
0
votes
2answers
101 views

Ackermann Steering Angle

Given the position of the vehicle ($x,y$) at different time points, the speed of the vehicle (m/s), the direction the vehicle is facing (heading — in degrees), the track width of the vehicle, and the ...
1
vote
1answer
30 views

Proving that motion of an $n$ dimensional oscillator can be written as a linear combination of “sine waves”

Here is a related question which might provide some context: LINK. Let's consider an oscillator with equation of motion in $n$ dimensions: $$ \frac{d^2}{dt^2} \vec{x} = K \vec{x}. $$ Given that $\...
0
votes
0answers
39 views

Orthonormalization of eigenamplitudes

Assuming $(-\omega^2 \hat m + \hat k)\vec{a}=0$ where $\vec a$ is the eigenamplitude of the eigenfrequency $\omega$ , $\hat m$ is the mass matrix and $\hat k$ is the matrix of the potential constants. ...
2
votes
2answers
128 views

Why must momentum operator in infinite well be self adjoint?

First, let me preface this statement by saying I know that there exists no (unique) self adjoint extension of the standard differential operator for the space $L_2([0,1])$. However, when one attempts ...
-2
votes
1answer
64 views

What are the Eigenstates and Eigenvalues? [closed]

In quantum mechanics I keep hearing about them. Kindly tell about them...not at a very very high level but simple enough to understand completely
0
votes
0answers
35 views

Coupled Harmonic Oscillator (Forced Vibration)

I derived two equations for a 2DOF harmonic oscillator system, declared state variable equations, and placed them into matrix form: $Ax' + Bx = C$. I have a Matlab script to determine the constants ($...
0
votes
0answers
34 views

Order of positions of tensor/vector components in an inner/outer product

Show that if $T_i$ are the components of covariant vector T, then $S_{ij}=T_iT_j-T_jT_i$ are the components of a skew-symmetric covariant tensor S. The question is whenever working with equations of ...
1
vote
0answers
47 views

How to do Weierstrass-transform in MATLAB? [closed]

I have a diagonalization problem. I have the eigenstates correctly, and I want to do a Gaussian-smearing (Weierstrass-transform?) on them. So I have the wave functions ($\Psi$, $1\times N$ vectors), ...
0
votes
1answer
37 views

Complex conjugate in inner products [duplicate]

When we solve for inner product of $\rvert a \rangle \cdot \rvert b \rangle$ we solve for $\langle a \rvert b \rangle$ where $\langle a \rvert$ is complex conjugate of $\rvert a \rangle$. However this ...
0
votes
1answer
1k views

Inversion of a metric

I am currently reading a paper by Bredberg $et.al$ arXiv:1101.2451 titled "From Navier-Stokes to Einstein". In this paper, the authors have considered a metric of the form \begin{eqnarray}ds^2_{p+2} = ...
0
votes
0answers
31 views

Sling loads for multipoint lifts

I am trying to calculate sling loads for a n-point lift. I want to utilize vector calculations and make it as general as possible, and also work in 3D-space. The idea is to use position vectors for ...
0
votes
1answer
26 views

Prove acceleration in orbit with newtons second law

I want to prove that the acceleration in a orbit at a given point r=(x,y) is $a=-\frac{GM}{R^3}r$ (My professor said this can be proven by newtons second law, but he never explained in detail how). I ...
0
votes
1answer
46 views

Question on notation for the inner product of complex vectors [duplicate]

Regarding the wiki: https://en.wikipedia.org/wiki/Sesquilinear_form#Hermitian_form you can see that the wiki states that physics defines the inner product for complex vectors as: $$\langle \, \...
0
votes
1answer
36 views

Can someone please explain the meaning of the circled paragraph?

why does the off diognal elements of the matrix mediate with the coupling differential equation?
0
votes
2answers
59 views

How to prove $(α·σ)(β·σ) = α·β +iα×β·σ$ (where, $α$ and $β$ are 3 dimensional vectors and $σ$ represents Pauli matrices)?

I tried to evaluate the LHS first and obtained the first term of RHS easily. Then i tried to use the commutation relations of $\mathrm{SU}(2)$ group to proceed further to obtain the second term of the ...
16
votes
8answers
3k views

Formal Definition of Dot Product

In most textbooks, dot product between two vectors is defined as: $$\langle x_1,x_2,x_3\rangle \cdot \langle y_1,y_2,y_3\rangle = x_1 y_1 + x_2 y_2 + x_3 y _3$$ I understand how this definition ...
5
votes
1answer
305 views

Is a vector space automatically spacelike if it has a basis of spacelike vectors?

I am studying Kerr Spacetime and I am not sure about something used in a proof I am trying to understand. I am wondering, if you consider a 4-dimensional Lorentzian manifold $\mathcal{M}$ and $X_i \...
0
votes
0answers
20 views

Normal mode decomposition of a triangular hexagonal lattice

I was trying to understand and redo the methods used in a previous question: Vibrational anharmonic coupling and noise-induced spontaneous symmetry breaking in a hexagonal finite mechanical lattice ...
0
votes
1answer
79 views

Why are coherent states not linearly independent?

From the completeness relation one can see that, $$|\psi \rangle = \int \frac{d^2 \alpha}\pi \langle \alpha | \psi \rangle |\alpha\rangle.$$ And if $|\psi\rangle = |\beta \rangle$ (which is another ...
0
votes
0answers
28 views

Math of anyons: Quantum dimension of 1 implies abelian charge

This question originates from the following statement in Bonderson's thesis: Link to Thesis page 16 or pdf-page 23: The quantum dimension $d_a$ of an anyon of charge $a$ satisfies $d_a \geq 1$ with ...
0
votes
1answer
59 views

Linear and angular speeds of a train

I am using Radar sensor in a train. Sensor is in the front part of the train to detect object and avoid collision. It needs vehicle motion data to calculate objects longitudional and lateral speeds....
0
votes
0answers
49 views

How to understand notation in “Introduction to Quantum Mechanics (3rd Edition)” by David Griffiths, Chapter 3.6.2?

In the 3rd edition, on page 118, the projection operator is introduced as $$\hat{P}=|\alpha\rangle\langle\alpha|.$$ Then Griffiths says that when $\hat{P}$ acts on another vector, it looks like ...
1
vote
1answer
70 views

How can quantum operators be expressed as a matrices?

I have just started quantum mechanics with Shankar. In my understanding, quantum operators are linear operators in infinite-dimensional Hilbert spaces. Shankar has repeatedly treated quantum ...
1
vote
0answers
34 views

How contravarient metric tensor equals the cofactor of covarient metric tensor over its determinant? [closed]

Is the contravariant form of the metric tensor the inverse of the covariant form of the metric tensor?
1
vote
1answer
65 views

Microcausality when quantizing the real scalar field with anticommutators

We know by the spin-statistics theorem that the real scalar field has to be canonically quantized by commutators. But if we try to use anticommutators, we would expand the field $$\phi(x)=\int\frac{d^...
1
vote
2answers
74 views

Exact solution for the perturbation of the inverse metric

So when we usually linearize general relativity with respect to metric perturbations $g_{\mu\nu}\rightarrow g_{\mu\nu}+h_{\mu\nu}$, we compute the correction to the inverse of the metric to first ...
10
votes
4answers
782 views

0-rank tensor vs vector in 1D

What is the difference between zero-rank tensor $x$ (scalar) and vector $[x]$ in 1D? As far as I understand tensor is anything which can be measured and different measures can be transformed into ...
3
votes
1answer
107 views

Is any given triplet spin state an eigenstate of some $j^z$ in the suitable basis?

Imagine you have a triplet spin state, which, in general, can be written as $$|\psi \rangle = \alpha | \uparrow \uparrow \rangle + \beta ( | \downarrow \uparrow \rangle+ | \uparrow \downarrow \...
0
votes
1answer
33 views

Is a dichotomic basis possible for 3-dimensional space?

We know that the Pauli basis for the 2-dimensional space is a dichotomic basis in the sense that every Pauli matrix has two distinct eigenvalues. Is it possible to express a 3-dimensional matrix $\...