# 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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### Need help understanding matrix representation of a linear operator

I'm struggling with linear algebra. Specifically, understanding the following: $\newcommand{\ket}{|#1\rangle}$ Suppose $A:V \rightarrow W$ is a linear operator between vector spaces $V$ and $W$. ...
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### Why are galilean transforms affine? [duplicate]

Here is a decomposition of galilean transforms of the form $x\mapsto Ax+y.$ Why are they all of this form? $T$ galilean is distance preserving so it is also injective. Take $B_r(a)$ the closed $r-$...
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### What are the principal applications of linear algebra in theoretical physics? [closed]

I am currently doing Shankar's Principle of Quantum Mechanics, and I am wondering besides the Quantum Mechanics applications that I already know it's a myriad, what are the other things inside physics ...
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### In order to Schwarz's Inequality becomes an equality, why is it necessary to the vectors be parallel or antiparallel?

Consider the $n$-dimensional vectors $|V\rangle$ and $|W\rangle$. Recall Schwarz's Inequality: $$|\langle V|W\rangle| \leqslant |V||W|$$ I want to understand why the only way to the equality hold ...
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### Is it worth to understand all linear algebra abstract proofs in order to understand QM? [closed]

I am studying quantum mechanics from Shankar's book, Principles Of Quantum Mechanics, and I started from the very first chapter because he makes the necessary mathematics introduction with linear ...
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### Constructing the concept of a tensor from simpler, more familiar ideas

I am having significant difficulty understanding where tensors fit into the big picture. I would really like to see exactly how tensors relate to simpler concepts that I am familiar with such as ...
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### Can we replace our use of cross products with the BAC-CAB rule?

A trending question here asks a question about cross products which are related to the existence of an orientation on 3D space. At some level what we are trying to do with cross products seems to ...
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### If a basis set is complete, are the elements in it mutually orthonormal?

If a basis set is complete, are the elements in it mutually orthonormal? For example, we can express the field operator in the basis of the creation and annihilation operators.This basis is complete, ...
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### Generalised dot and cross products

Are the dot and cross products in $\Bbb R^2$ and $\Bbb R^3$ specific examples of a more general operation that is defined on arbitrary dimensional vector spaces? And if so are there any physical ...
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### Does the geometric algebra of curved space have a matrix representation?

Suppose the geometric algebra defined by $$\frac{1}{2}(e_\mu e_\nu +e_\nu e_\mu)=g_{\mu\nu}$$ where $e_\mu,e_\nu$ are generators of the algebra, and where $g_{\mu\nu}$ are elements of the reals. I ...
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### 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 ...
From a coordinate-free point of view, we can rewrite the Einstein Field Equations $R^{\mu}\hspace{0.5mm}_{\nu} - \frac{1}{2}R \delta^{\mu}\hspace{0.5mm}_{\nu}=:G^{\mu}\hspace{0.5mm}_{\nu} = 8\pi T^{\... 1answer 37 views ### Does the symmetrization of the wave function change the energy? Suppose two non interacting electrons, in a time independent potential, described by the equation: \begin{equation} {H} \psi(r_1, r_2) = \frac{-\hbar^2}{2m} (\nabla^2_1 + \nabla_2^2) \psi(r_1, r_2) + ... 1answer 89 views ### What is the difference between a dual vector and a reciprocal vector? I am familiar with the concept of a dual space$V^*$as the set of all linear functionals$\tilde{\omega}: V \rightarrow \mathbb{R}$. The inner product on$V$is usually used to define the dual of a ... 2answers 73 views ### Differentiation of the determinant$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}$is a cofactor, but I can't ... 3answers 170 views ### A form$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$is ... 0answers 27 views ### Hermitian phase operator and quantum harmonic oscillator I need to apply a hermitian phase operator$\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 ... 0answers 42 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? 1answer 59 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, ... 0answers 11 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. ... 3answers 86 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? ... 0answers 20 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 ... 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 ... 5answers 590 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 ... 1answer 31 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 $$[x,y]=\langle x,gy\... 0answers 33 views ### Proof of skewsymmetry of electromagntic function in Minkowski spacetime I have been studying special relativity from the Gregory Naber's book: "The geometry of Minkowski spacetime" and I found a very strange proof. In Section 2.1, just before of equation 2.1.2. the book ... 1answer 94 views ### Reading energy Eigenvalues from a Hamiltonian matrix for 1D harmonic oscillator After a perturbation V(x) added to the system, a matrix element H_{nn} calculated in unperturbed Eigenstates for one-dimensional harmonic oscillator is given as:$$\epsilon \hbar \omega_0\begin{... 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 ... 1answer 51 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 ... 2answers 352 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 ... 1answer 55 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), $$S_1 = s \sin \... 1answer 54 views ### Understanding completeness relation and writing Hamiltonian in matrix form A three level system hamiltonian I found where it is written as:$$\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^* |... 1answer 48 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$\...
I would like to prove this relation: $$g^{\mu\nu} = \frac{1}{3!} \frac{1}{g} \epsilon^{\mu\rho\sigma\kappa}\epsilon^{\nu\alpha\beta\gamma} g_{\rho\alpha} g_{\sigma\beta} g_{\kappa\gamma}, \tag{1}$$ ...