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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Understanding the inverse in the definition $(\tilde{\Pi}_gf)(v)\equiv f(\Pi^{-1}_gv)$

I'm trying to understand the representation $\tilde{\Pi}$ induced from the fundamental representation $\Pi$, defined as $(\tilde{\Pi}_gf)(v)\equiv f(\Pi^{-1}_gv)$ for $g\in G,\hspace{1mm}f\in\mathcal{...
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Looking for a freeware software/app that can solve eigenvalue problems symbolically

I'm taking a quantum mechanics course and my homework involves extremely tedious algebra to solve symbolic eigenvalue problems. I'm looking for a software that I can give matrices with symbolic ...
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33 views

How to prove that all eigenstate $|0\rangle,|1 \rangle,…| n\rangle $ are non-degenerate? [duplicate]

How to prove that all eigenstate $|0\rangle,|1 \rangle,...| n\rangle $ are non-degenerate where $a^{\dagger}a|n \rangle =n|n \rangle $
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Commutator of $B$, $C$ vanishes if $A$, $B$, $C$, $AB$, $AC$ are Hermitian

Suppose 3 operators $A$, $B$, $C$ are Hermitian operators. Assume $A$ has a non-degenerate spectrum, and $AB$ and $AC$ are also Hermitian. Show that $$[B,C] = 0$$ From the conditions $A$, $B$, $C$, $...
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Need help understanding matrix representation of a linear operator

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

4×4 Cofactor Transpose Matrix calculation gone wrong, in Shankar's Principles of Quantum Mechanics

In Appendix A$.1$, Shankar, R; Principles of Quantum Mechanics, the cofactor transpose of a $3\times3$ matrix $M$ is given as (to be referred to as the first procedure by me) $$\overline{M}=\begin{...
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90 views

Congruence transformations of matrices

From the book Analytical Mechanics by Fowles and Cassiday I am studying classical coupled harmonic oscillators. These are systems that are governed by a system of linear second order differential ...
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How am I to interpret $\text{Tr}(\text{ad}_X\text{ad}_Y)$?

I'm trying to show that the $(2,0)$ Killing tensor is invariant under the $\text{Ad}$ homomorphism: $K(\text{Ad}_A(X),\text{Ad}_A(Y))=K(X,Y),$ with $X,Y\in \mathfrak{g},\hspace{1mm}A\in G,$ and $K(X,Y)...
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32 views

How to demonstrate that Fierz-like identity for 2-components Weyl spinors? [duplicate]

Consider the 2-components Weyl spinors with the following scalar product \begin{equation}\tag{1} \langle \, \phi, \, \psi \, \rangle = \phi^{\top} \, \sigma_y \: \phi, \end{equation} where $\sigma_y$ ...
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44 views

Simultaneous diagonalization of Cartan generators of $SO(6)$

This question is naive but for some reason I'm not getting the expected result. The generators of $SO(6)$ can be written in this way: $$(J_{ab})_{cd}=i(\delta_{ac}\delta_{bd}-\delta_{ad}\delta_{bc}),...
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22 views

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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1answer
52 views

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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62 views

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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60 views

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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1answer
66 views

Hilbert space operators in $V⊗V^\dagger$ and $V⊗V$?

Every operator in Hilbert space is defined in space $V⊗V^\dagger$ respectively $$A=\sum_{ij}a_{ij}|i\rangle\langle j|$$ But identical operator when we can defined in space $V⊗V$ as $$A=\sum_{ij}a_{...
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30 views

Schmidt decomposition of bipartite states

When performing the Schmidt decomposition of a bipartite state $\left|\psi\right>_{AB}=\sum_{ij}c_{ij}\left|v_i\right>_{A}\left|w_j\right>_{B}$ is computing the eigenvectors of the reduced ...
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283 views

Getting the arbitrary boost matrix from a similarity transformation

Note: For the following question I'm using the non-standard $(x,y,z,ct)$ notation. I'm wanting to represent an arbitrary boost in the $\hat{\beta}$ direction by doing a similarity transformation on ...
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87 views

The diagonal representation of Pauli $y$ matrix?

The textbook says the eigenvalue of the Pauli y matrix is 1 and -1, the corresponding eigenvectors are, $$\sqrt{\frac{1}{2}} \begin{bmatrix} 1\\ i \end{bmatrix} , \sqrt{\frac{1}{2}} \begin{bmatrix} 1\...
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2answers
76 views

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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1answer
58 views

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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1answer
48 views

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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1answer
61 views

Misconception in partial derivatives of Lorentz transformation

Let us consider a Lorentz transformation of four vectors from frame S to S' where S' is moving with relative velocity $\textbf{v}$ with respect to S. The boost is given by $$t'=\gamma(t-vx), \quad x'=\...
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Is point-inversion in the bloch sphere a valid quantum channel?

Consider the qubit map $\mathcal{P}: \mathcal{P}(|\eta\rangle\langle\eta|) = |\bar{\eta}\rangle\langle\bar{\eta}|$, where $\langle\bar{\eta}|\eta\rangle = 0$. On the Bloch sphere, $\mathcal{P}$ ...
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Spectral form factor for integrable systems

The spectral form factor is defined as the Fourier transform of the autocorrelation function of the energy density $$K(t) = \frac{\int d E_1 dE_2 \langle \rho(E_1) \rho(E_2)\rangle e^{i(E_1 - E_2)t}}{...
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1answer
68 views

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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1answer
95 views

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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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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37 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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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) + ...
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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 ...
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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 ...
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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 ...
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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 ...
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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?
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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. ...
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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? ...
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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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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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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 ...
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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\...
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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 ...
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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{...
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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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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 ...
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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 ...
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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 \...
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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^* |...
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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 $\...
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1answer
76 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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