Questions tagged [linear-algebra]

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

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Basis of the Lie Algebra of a Group [migrated]

It is known that the Lie algebra of a group is a vector space. The question i have is this: Is there a way to find a basis of the Lie algebra of the group? Also, if i have a set of matrices that ...
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1answer
23 views

Generalization to Rear-Wheel Steering?

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 ...
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33 views

Doubt about the “kernel of Einstein's equations”

From a coordinate-free point of view, we can rewrite the Einstein Field Equations $R^{\mu}\hspace{0.5mm}_{\nu} - \frac{1}{2}R \delta^{\mu}\hspace{0.5mm}_{\nu}=:G^{\mu}\hspace{0.5mm}_{\nu} = 8\pi T^{\...
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Does the symmetrization of the wave function change the energy?

Suppose two non interacting electrons, in a time independent potential, described by the equation: \begin{equation} {H} \psi(r_1, r_2) = \frac{-\hbar^2}{2m} (\nabla^2_1 + \nabla_2^2) \psi(r_1, r_2) + ...
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1answer
192 views

Eigenvalues of the Klein-Gordon operator

If I've understood what I've read correctly, the eigenvalues of the Klein-Gordon (KG) operator $\Box+m^{2}$ are $-p^{2}+m^{2}$, but how does one show this? Naively I assumed that the eigenfunctions ...
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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} = ...
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72 views

What is the difference between a dual vector and a reciprocal vector?

I am familiar with the concept of a dual space $V^*$ as the set of all linear functionals $\tilde{\omega}: V \rightarrow \mathbb{R}$. The inner product on $V$ is usually used to define the dual of a ...
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234 views

What the primed quantities really are in this context?

In this definition: The set of $N$ quantities $V_j$ is said to be the components of an $N$-dimensional vector $\mathbf{V}$ if and only if their values relative to the rotated coordinate axes are ...
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3answers
160 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 ...
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2answers
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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 ...
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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 ...
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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?
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308 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 \...
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46 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, ...
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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. ...
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3answers
79 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? ...
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140 views

Uniqueness of simultaneous eigenstates of two linear operators

I was solving a homework problem where the question gives the representation of two operators in matrix form, in some arbitrary set of basis vectors. It then asks to find the simultaneous eigenstates ...
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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 ...
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2answers
154 views

Why do the density operators span the whole operator space $\mathcal{B}(H)$?

The convex set of density operators on a finite-dimensional Hilbert space $H$ defined by $$\mathcal{D}(H):=\{\rho\in\mathcal{B}(H)\,|\,\rho\geq 0,\, \operatorname{tr}\rho =1\},$$ This set is said to ...
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4answers
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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 ...
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2answers
342 views

Why is the dimension of the set separable states $\dim\mathcal H_1+\dim\mathcal H_2$?

Please can you help me to understand how the dimension of the set of separable states is $\dim \cal H_1 + \dim \cal H_2$? This is the relevant passage: So far, we have assumed implicitly that the ...
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5answers
487 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 ...
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1answer
49 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 \...
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4answers
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How do you find a particular representation for Grassmann numbers?

This question is more general in the sense that I want to know how one finds a particular (say matrix) representation for any object. For the case of Grassmann numbers we have from Wikipedia the ...
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1answer
29 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\...
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2answers
467 views

Determine the point at which moment vector is zero on a 3D body

I have information about total force and moment on a body for three points, whose coordinates I know. From this information I would like to determine the point at which moment would be zero. ...
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2answers
653 views

Vectors ( Resolution of vectors )

How many components can a vector be resolved into? I think that it should be infinity because there can be infinite axes. Am I right?
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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 ...
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1answer
496 views

Fermionic statictics in $SU(2)$ slave-boson representation

One of the $SU(2)$ slave-boson decompositions has been introduced by X.-G. Wen and P. A. Lee in PRL, 76, 503 (1996). (A generic recipe for constructing the SU(2) slave-particle framework has been ...
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60 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{...
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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 ...
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2answers
137 views

Determinant of ADM metric

I am studying inflation and for the calculation of the bispectrum we are using the ADM formalism where the metric is the following form: $$g_{\mu\nu}=\begin{bmatrix}-N^2+N^iN_i&N_i\\N_i&h_{ij}...
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1answer
33 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 ...
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2answers
337 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 ...
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1answer
528 views

What's the physical meaning of the kernel of density matrix?

The kernel of this linear map is the set of solutions to the equation A x = 0, where 0 is understood as the zero vector. But what's the physical meaning of the kernel of density matrix?
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1answer
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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^* |...
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Are matrices and second rank tensors the same thing?

Tensors are mathematical objects that are needed in physics to define certain quantities. I have a couple of questions regarding them that need to be clarified: Are matrices and second rank tensors ...
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1answer
35 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 $\...
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1answer
69 views

Identity for the inverse metric tensor using its determinant

I would like to prove this relation: $$g^{\mu\nu} = \frac{1}{3!} \frac{1}{g} \epsilon^{\mu\rho\sigma\kappa}\epsilon^{\nu\alpha\beta\gamma} g_{\rho\alpha} g_{\sigma\beta} g_{\kappa\gamma}, \tag{1}$$ ...
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1answer
43 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=...
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2answers
131 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
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1answer
244 views

Finding basis of Schmidt decomposition

How do you find the basis for the Schmidt decomposition when given a state of multiple qubits? For example, if you have the systems $A,B$ and $C$, how do they correspond to eigenvectors of a density ...
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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 $\...
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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. ...
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2answers
134 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 ...
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1answer
68 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
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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 ($...
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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 ...
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1answer
39 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 ...
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33 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 ...