Questions tagged [linear-algebra]
To be used for linear algebra, and closely related disciplines such as tensor algebras and maybe clifford algebras.
525
questions
1
vote
2answers
43 views
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{...
0
votes
0answers
25 views
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 ...
0
votes
0answers
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 $
1
vote
1answer
74 views
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 $A$, $B$, $C$, $...
0
votes
2answers
64 views
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 $W$. ...
0
votes
0answers
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)
$$\overline{M}=\begin{...
1
vote
2answers
90 views
Congruence transformations of matrices
From the book Analytical Mechanics by Fowles and Cassiday I am studying classical coupled harmonic oscillators. These are systems that are governed by a system of linear second order differential ...
1
vote
3answers
78 views
How am I to interpret $\text{Tr}(\text{ad}_X\text{ad}_Y)$?
I'm trying to show that the $(2,0)$ Killing tensor is invariant under the $\text{Ad}$ homomorphism: $K(\text{Ad}_A(X),\text{Ad}_A(Y))=K(X,Y),$ with $X,Y\in \mathfrak{g},\hspace{1mm}A\in G,$ and $K(X,Y)...
2
votes
0answers
32 views
How to demonstrate that Fierz-like identity for 2-components Weyl spinors? [duplicate]
Consider the 2-components Weyl spinors with the following scalar product
\begin{equation}\tag{1}
\langle \, \phi, \, \psi \, \rangle = \phi^{\top} \, \sigma_y \: \phi,
\end{equation}
where $\sigma_y$ ...
0
votes
1answer
44 views
Simultaneous diagonalization of Cartan generators of $SO(6)$
This question is naive but for some reason I'm not getting the expected result.
The generators of $SO(6)$ can be written in this way:
$$(J_{ab})_{cd}=i(\delta_{ac}\delta_{bd}-\delta_{ad}\delta_{bc}),...
1
vote
0answers
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 $r-$...
1
vote
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 ...
-1
votes
1answer
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 ...
1
vote
0answers
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 ...
0
votes
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
$$A=\sum_{ij}a_{...
1
vote
0answers
30 views
Schmidt decomposition of bipartite states
When performing the Schmidt decomposition of a bipartite state $\left|\psi\right>_{AB}=\sum_{ij}c_{ij}\left|v_i\right>_{A}\left|w_j\right>_{B}$ is computing the eigenvectors of the reduced ...
1
vote
2answers
283 views
Getting the arbitrary boost matrix from a similarity transformation
Note: For the following question I'm using the non-standard $(x,y,z,ct)$ notation.
I'm wanting to represent an arbitrary boost in the $\hat{\beta}$ direction by doing a similarity transformation on ...
0
votes
2answers
87 views
The diagonal representation of Pauli $y$ matrix?
The textbook says the eigenvalue of the Pauli y matrix is 1 and -1, the corresponding eigenvectors are,
$$\sqrt{\frac{1}{2}}
\begin{bmatrix}
1\\
i
\end{bmatrix}
, \sqrt{\frac{1}{2}}
\begin{bmatrix}
1\...
1
vote
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 ...
0
votes
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 ...
0
votes
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, ...
0
votes
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
$$t'=\gamma(t-vx), \quad x'=\...
0
votes
0answers
22 views
Is point-inversion in the bloch sphere a valid quantum channel?
Consider the qubit map $\mathcal{P}: \mathcal{P}(|\eta\rangle\langle\eta|) = |\bar{\eta}\rangle\langle\bar{\eta}|$, where $\langle\bar{\eta}|\eta\rangle = 0$. On the Bloch sphere, $\mathcal{P}$ ...
0
votes
0answers
18 views
Spectral form factor for integrable systems
The spectral form factor is defined as the Fourier transform of the autocorrelation function of the energy density
$$K(t) = \frac{\int d E_1 dE_2 \langle \rho(E_1) \rho(E_2)\rangle e^{i(E_1 - E_2)t}}{...
0
votes
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 ...
2
votes
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 ...
0
votes
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 ...
0
votes
0answers
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^{\...
0
votes
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) + ...
0
votes
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 ...
3
votes
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 ...
1
vote
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 ...
0
votes
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 ...
0
votes
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?
0
votes
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, ...
0
votes
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. ...
0
votes
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?
...
0
votes
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 ...
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
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 ...
2
votes
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 $$[x,y]=\langle x,gy\...
0
votes
0answers
33 views
Proof of skewsymmetry of electromagntic function in Minkowski spacetime
I have been studying special relativity from the Gregory Naber's book: "The geometry of Minkowski spacetime" and I found a very strange proof. In Section 2.1, just before of equation 2.1.2. the book ...
0
votes
1answer
94 views
Reading energy Eigenvalues from a Hamiltonian matrix for 1D harmonic oscillator
After a perturbation $V(x)$ added to the system, a matrix element $H_{nn}$ calculated in unperturbed Eigenstates for one-dimensional harmonic oscillator is given as:
$$\epsilon \hbar \omega_0\begin{...
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
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 ...
4
votes
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 ...
2
votes
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),
$$ S_1 = s \sin \...
0
votes
1answer
54 views
Understanding completeness relation and writing Hamiltonian in matrix form
A three level system hamiltonian I found where it is written as:
$$\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^* |...
0
votes
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 $\...
1
vote
1answer
76 views
Identity for the inverse metric tensor using its determinant
I would like to prove this relation:
$$g^{\mu\nu} = \frac{1}{3!} \frac{1}{g} \epsilon^{\mu\rho\sigma\kappa}\epsilon^{\nu\alpha\beta\gamma} g_{\rho\alpha} g_{\sigma\beta} g_{\kappa\gamma}, \tag{1}$$
...
