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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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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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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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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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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. ...
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? ...
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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 ...
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 ...
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 ...
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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 ...
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 ...
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 ...
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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 ...
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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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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), ...
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 ...
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} = ...
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 ...
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 ...
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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 ...
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 ...
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 \...