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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Position of a point on a rigid body

I can not wrap my head around the following doubt: How can we express (or prove the fact that) the position of a material point on a rigid body as the sum of a "traslational component" and ...
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Finding the eigenvalues and eigenvectors of Pauli matrices [closed]

Pauli's matrices are $\hat\sigma_x =\left(\begin{array}{cc} 0 & 1\\ 1 & 0\\ \end{array}\right) $ $\hat\sigma_y =\left(\begin{array}{cc} 0 & -i\\ i & 0\\ \end{array}\right) $ $\hat\...
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Is there a space associated with a collection of functions which holds all consequences of said functions as objects?

I am an undergraduate student studying physics. I am working through a linear algebra book this summer to deepen my understanding, so I am thinking generally about organizing mathematical objects. I ...
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3 votes
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Eigenkets of a two-state hamiltonian

I have a question related to this other question: Eigenenergies and eigenkets given the Hamiltonian. In it, OP is given the following hamiltonian: $$ H=a(|1\rangle \langle1|-|2\rangle\langle2|+|1\...
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Why are Eigenvectors of a 1D quantum ising hamiltonian real

I was modelling the 1D transverse quantum Ising model and made a Kronecker product loop to find the Hamiltonian of the system, for a given magnetic field configuration. Now, my question is that when I ...
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2 votes
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Measurement operator in a Bell experiment

I'm trying to figure out why a Bell experiment gives rise to the payoff (measurement) operator used in this paper on quantum game theory. Two players are each in control of one half of an entangled ...
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Special relativity book which describes concepts using linear algebra notions

It seems so every idea of special relativity can be formulated quite nicely in Linear algebra notions such as the inner product matrix and change of basis matrices. However, I can't find a single book ...
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Is Singularity corresponds to null space and are forces causing singularity are some transformation matrix?

Some random thoughts: (Disclaimer: i am a complete noob in physics). I was studying linear transformations from linear algebra and just got this thought that there might be some set of transformation ...
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Whats the meaning of the 1 Ket? [closed]

I am talking this one: $|1\rangle$. If I have 2 orthonormal states $|1\rangle$ and $|2\rangle$ in the 2D Hilbert space, does that imply the vector $\vec{\psi_n}=(1,2)$, if I would like to solve the ...
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2 votes
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Equation for the simultaneous eigenfunction of three operators in spherical coordinates

If I'm considering the three operators $H,L^2,L_z$ with the condition $[H,L^2]=[H,L_z]=[L^2,L_z]=0$, I can find a complete set of simultaneous eigenfunctions. If I study this problem in spherical ...
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How the eigenvalue problem was solved?

In Gasiorowicz 3rd edition Chapter 3, I've tried to solve this problem I checked the solution's manual, When I tried to integrate it, the answer I got is $$ \psi(x)=Ce^{x^2/2\lambda} $$ Can you ...
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Factorization of the wavefunction in a central Hamiltonian problem

I am trying to understand the topic of the title. If I consider a central Hamiltonian, so an Hamiltonian of the form $H=\frac{p^2}{2m}+V(r)$ what are the logical steps that lead me to the known result?...
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Quantum mechanics, are simultaneous eigenstates to be intended always as a tensorial product of two eigenstates?

The question is the one of the title, let $\hat{O}_1$ and $\hat{O}_2$ two commuting operators: $[\hat{O}_1,\hat{O}_2]=0$, there is an orthonormal basis formed by their simultaneous eigenstates. These ...
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What is an eigensystem? Could you provide a simple example? [closed]

Also, what is the difference between an eigensystem and the eigenspace?
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Force to Inflate a ball underwater [closed]

How much force is required to fully inflate (with air) a beach ball that is 6 feet in diameter at depths of 200 feet underwater?
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Geometric multiplicity vs degree of degeneracy of energy levels

Let $A$ be a square matrix of order $n$. Given a fixed eigenvalue $\lambda$, we call geometric multiplicty the dimension of the associated eigenspace $$g_{\lambda} = \dim(v\in\mathbb{K}^n : Av= \...
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Are there ways to find representations of matrices given an algebra?

Given an equation (or a set of equations) involving matrices, is there an algorithm to find possible representations of these matrices? For example, we can consider a matrix $A$ such that $A^2=\begin{...
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Is the Lindblad equation invariant under the unitary transformation?

Let us say we have a Lindblad equation for the density matrix $$ \dot{\rho}=-\frac{i}{\hbar}[H, \rho]+\sum_{i=1}^{N^{2}-1} \gamma_{i}\left(L_{i} \rho L_{i}^{\dagger}-\frac{1}{2}\left\{L_{i}^{\dagger} ...
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3 votes
2 answers
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How to write a matrix $\mathcal{M}$ such that $\mathcal{M} \boldsymbol{x}=\boldsymbol{\omega}\times\boldsymbol{x}$? [duplicate]

As is well known, it is possible to use the $\nabla$ operator as if it were a vector.  Someone consider it an abuse of notation but surely something that works well and is very useful. Well, how is it ...
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Energy (Hamiltonian) of Trial Wavefunction

Here I give a part of derivation of Hartree-Fock equations in case where basis functions (wavefunctions) are orthonormal and real: $$ \langle \psi_i | \psi_j \rangle = \langle \psi_j | \psi_i \rangle =...
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Square of Volume of Tetrahedron in Loop Quantum Gravity

In the subsequent statement about the geometry of Tetrahedron in Carlo Rovelli's book, there is a formula as follows: $$V^2 = \frac {2}{9}(\vec L_1 \times \vec L_2)\cdot \vec L_3$$ Where $\vec L_a$ is ...
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Please explain statement in a book on Loop Quantum Gravity

In a book by Carlo Rovelli on Covariant Loop Quantum Gravity, I struggle to understand a statement on Tetrahedron as follows: What is the dimension of the matrix? How to derive the given matrix ...
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Forming an upper bound [duplicate]

\begin{eqnarray*} \|\left(E_{i} \otimes I_{d}\right)|\psi\rangle-\left(I_{d} \otimes F_{i}\right)|\psi\rangle \|^2&=&\|\left(E_{i} \otimes I_{d}\right)|\psi\rangle\|^2-2\langle\psi|E_i\otimes ...
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Pure and mixed density operators of a Schmidt decomposition

Suppose we have Hilbert space factorisable in to K subsystems $$ \mathcal{H} = \mathcal{H}_1\otimes...\otimes\mathcal{H}_K $$ in which we can express a pure state as the Schmidt decomposition $$ |\...
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Can the time-evolution operator be factorised if the Hamiltonian is a sum of two commuting operators?

Let the time-dependent Hamiltonian $H(t) = A(t) + B(t)$ for some quantum system be given as the sum of two time-dependent operators $A(t)$ and $B(t)$. Further, assume that $A(t)$ and $B(t)$ commute, ...
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$y$ Pauli Operators Eigenvectors - How are they orthogonal?

I am struggling to obtain that the eigenvectors of the Pauli $y$ operator are orthogonal, and would appreciate guidance on where I am going wrong. I have calculated the eigenvalues as: 1, -1 And ...
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What does eigenvalues of the Lorentz matrix represent physically speaking? [duplicate]

In special relativity, if we have a boozt in the x - direction, the relationship between the coordinates of the inertial frame of reference S, and the one of S' (moving with velocity v relative to S), ...
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Which of these is the logical way to establish tensors on a manifold?

You start by defining a vector space at each point of the manifold. The defining feature being the vector transformation law under change of co-ordinates. Then you define dual vectors as linear ...
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Inner product evaluation in QM

On wikipedia on the page for inner product it states that for any two $x,y$ in a vector space $V$ the inner product $(\cdot , \cdot)$ satisfies $(ax, y) = a(x,y)$ where $a\in\mathbb{C}$. The inner ...
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Srednicki eq. (1.27): $\left\{\alpha^{j}, \alpha^{k}\right\}_{a b}=2 \delta^{j k} \delta_{a b}$

Srednicki, QFT, p. 8 writes $$\left\{\alpha^{j}, \alpha^{k}\right\}_{a b}=2 \delta^{j k} \delta_{a b}\tag{1.27}.$$ What does exactly $ab$ here denote? Assume I have a matrix X [0 1] [2 3] and does a ...
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2 votes
2 answers
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Minimum number of non-coplanar forces required to keep an object in equilibrium

The minimum number of non-coplanar forces that can keep a particle in equilibrium is: (a) 1 (b) 2 (c) 3 (d) 4 Answer given is option $(d)$ , i.e $4$. But can’t it be $(c)$ , i.e $3$ too? Suppose I ...
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Null vectors that aren't the zero vector in general relativity?

So I was trying to understand the null energy condition of $T_{μν}k^μk^ν≥0$ Where $k$ is an "arbitrary future-directed null vector" and couldn't really wrap my head around how the $k$ is ...
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-3 votes
1 answer
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Inner product question

Why is it valid to write that: $$\langle\alpha|a^{\dagger}\alpha\rangle=\langle a\alpha|\alpha\rangle$$ where $\alpha$ is the lowering operator, $\alpha^{\dagger}$ is the raising operator, and $a\in \...
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Finding $\theta$ and $\phi$ when qubit state is $\frac{1}{\sqrt 2}[i ,1]^T$

Because we know the state of a qubit can be described as: $$ |q\rangle=\cos{\frac{\theta}{2}}|0\rangle+e^{i\phi}\sin{\frac{\theta}{2}}|1\rangle\\\ \\ \theta, \phi \in \mathbb{R} $$ How do I find the ...
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Commutation relation confusion of ladder operators in Quantum Mechanics

Suppose that $X$ and $N$ are operators such that they follow the commutation relation $$[N,X]=cX$$ for some scalar c. In this Wikipedia article it is shown that if $|n \rangle$ is some eigenstate of ...
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Contracting lightlike vector with timelike/spacelike vectors

Working in GR (3+1 dimensions) with a certain metric, I would like to understand what result I get contracting a lightlike four-vector with a spacelike and/or a timelike four-vectors (I'm interested ...
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Operator on infinite dimensional Hilbert space: domain and range

Since a QM operator is a linear map, it is useful to think about them as functions. An operator $\hat A$ on a finite $N$-dimensional Hilbert space $H_N$ is always such that $$\hat A:H_N\to H_N.$$ The ...
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1 vote
1 answer
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Questions about Fock space and direct sum

I am a little bit confused with the concept of a Fock space and hope for some clarification. In general a Fock space seems to be constructed as the direct sum of $n$-particle Hilbert spaces. What ...
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Do the eigenvectors of different observables span the same Hilbert space?

The Hilbert space is spanned by independent bases. The textbook said that the eigenvectors of observable spans the Hilbert space. Do the eigenvectors of multiple observables span the same Hilbert ...
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3 answers
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Trouble proving properties of the density matrix

So let's say you've got a Hilbert space that's $n$-dimensional with some vector $\phi$, such that $\langle \phi | \phi \rangle = 1$; let $|\phi \rangle \langle \phi | = \rho$. Also say we've got some ...
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How flexible can a unitary transformation be?

Let's say you have a Hamiltonian with $k$ different terms, \begin{equation} \hat{H}=\sum_{i=1}^k a_k \hat{O}_k \end{equation} with real coefficients $a_k$ and Hermitian operators $\hat{O}_k$. Now, if ...
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Justifications for the index upper lower labels in tensor component transformations

A (1,0)-type tensor may be written as $$ V = V^{\mu} e_{\mu} $$ The component transforms as $$ V^{\nu} = A^{\nu}{}_{\mu^\prime} V^{\mu^\prime} $$ (the basis can transform similarly) My question is, ...
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What areas of research use linear algebra the most? [closed]

I'm in my 4th semester as a physics and recently added a minor in math. I'm in my second linear algebra course, and am finding it extremely interesting and fun to understand. To complete my minor, I ...
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5 votes
1 answer
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What is the point in using dual spaces for quantum mechanics?

I have been studying bra-ket notation for QM and have come across the concept of dual spaces and their relation to bra vectors. I can appreciate that the definition of a dual space is such that, given ...
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Trace of a linear operator in Dirac notation

I've been banging my head against a wall trying to find a proof for: $$Tr(𝑋) = ∑_𝑗⟨𝑗|𝑋|𝑗⟩.$$ This is supposedly fundamental knowledge. Can anyone help with the proof or direct me to a resource ...
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What is the physical meaning of the eigenvalues of a state-space representation of a physical system?

Consider the following state-space model of a physical system. $\begin{bmatrix} \dot{{x_1}} \\ \dot{{x_2}} \end{bmatrix} = \begin{bmatrix}\ 0 & 1 \\ 0 & -c/m \end{bmatrix}\begin{bmatrix} \ ...
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Why can the metric tensor always be diagonalized?

I'm reading through some general relativity notes. I have reached a part that I don't understand, probably because my linear algebra is not good enough. My questions relating to the image below are: ...
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2 votes
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Does quantum mechanics somehow generalize the concept of affine tensor?

From https://mathworld.wolfram.com/AffineTensor.html and https://encyclopediaofmath.org/wiki/Affine_tensor It seems affine tensor transforms via orthogonal matrices: $$A^{T} A = 1 $$ But, in quantum ...
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Show $I+\tau\mathcal{L}$ is completely positive when $\tau \leq 1/\lambda_{\mathrm{max}}(\mathcal{L})$

I am not very well-versed when in comes to open quantum systems which is why I need some help. In a paper, I encountered the following situation: Let $\mathcal{L}$ be a Lindbladian so the time ...
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