Questions tagged [linear-algebra]
To be used for linear algebra, and closely related disciplines such as tensor algebras and maybe clifford algebras.
1,112 questions
5votes
0answers
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Diagonalizing a permutation-invariant operator
Consider a system of $L$ qubits and an operator, $\mathcal{O}$, acting on the system. Every matrix element of this operator is nonzero in the computational basis, so there aren't any obvious conserved ...
- 911
4votes
1answer
168views
Non-equilibrium thermodynamics in an eigenvector basis
I am currently studying non-equilibrium thermodynamics in the linear regime, and I am wondering why we define the constitutive equations (i.e: the relationship between flux and force) in a basis that ...
- 2,164
0votes
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Can the similarity transformation of a finite group representation be made unitary? [migrated]
We know that any irrep of a finite group can be chosen to be unitary. Assume we have done so. Given a reducible unitary representation $U$, can the similarity transformation $S$ be chosen to be ...
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1vote
0answers
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How can groundwater spread be found using gradient descent?
Suppose that a droplet of water in an aquifer flow has the position, $\gamma_t \in \mathbb{R}^3$ with a time-independent hydraulic head $h(x,y)$ (a watertable elevation map), where, \begin{align*} ...
3votes
2answers
285views
The inverse of a specific metric tensor [closed]
I am studying general relativity and here is a problem I encountered: Suppose $$ \mathrm{d}s^2=-M^2(\mathrm{d}t-M_i\mathrm{d}x^i)(\mathrm{d}t-M_j\mathrm{d}x^j)+g_{ij}\mathrm{d}x^i\mathrm{d}x^j $$ or ...
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0votes
1answer
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Correlation between Gibbs rule and Linear Algebra?
Is there a direct connection between the Gibbs rule, which counts the degrees of freedom in an equilibrium system, and the concept of linear independence of equations in linear algebra? Because for ...
- 1
2votes
0answers
157views
Basis and inner product of the Dirac (Clifford) algebra
The Dirac algebra in 4D spacetime is composed of four $4\times 4$ gamma matrices $\{\gamma^\mu\}=\{\gamma^0,\gamma^1,\gamma^2,\gamma^3\}$ satisfying the following anticommutation relation: $$\{\gamma^\...
- 606
2votes
1answer
71views
Linear independence of POVM elements to prove information completeness
Assume we are working on a finite dimensional complex Euclidean space ${H}$ (a Hilbert space), so that we can safely use "operator" and "matrix" as synonyms. Define $\text{dim}(H) =...
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4votes
1answer
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The product of the square root of Pauli matrices
When we deal with the Dirac equation and its solutions, we often define the following quantities: $$ \sigma^\mu\equiv(\mathbf{1},\vec{\sigma}) \ \ \ \text{and} \ \ \ \bar\sigma^\mu\equiv(\mathbf{1},-\...
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1vote
2answers
101views
Why do we use matrix in physics? [closed]
I recently started learning theoretical physics by myself and my book (the theoretical minimum series), and in this book, hessian matrix is used for multivariable function, and i want to know that why ...
0votes
0answers
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How to use the symmetry to get the eigenstates in twisted bilayer graphene (TBG)?
In tripod model of twisted bilayer graphene (TBG), the Hamiltonian with $ \mathbf{k} = 0$ is \begin{align} H = \begin{pmatrix} 0 & T_{1} & T_{2} & T_{3} \\ T_{1} & -\mathbf{q}_{1} ...
2votes
2answers
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Why is momentum an "operator"?
Consider the 1D infinite square well problem (say of length $L$). The space of eigenfunctions all satisfy the boundary conditions that $\psi(0)=\psi(L)=0$ and the ground state wavefunction is given by ...
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3votes
1answer
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Operational meaning of spectral radius of quantum channels
It is known from Chap.6 of Wolf's lecture notes that CPTP maps or quantum channels have spectral radius 1. That is the absolute value of the eigenvalues is at most 1, corresponding to the asymptotic ...
- 205
8votes
5answers
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What is the direction of dual basis vectors relative to the natural basis vectors?
By natural basis vector I mean the one normally used when representing a vector, i.e. covariant and usually written with a subscript to indicate the co-ordinate. By dual basis vector I mean the ...
- 81
0votes
0answers
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Probability of measuring $S_z$ when the particle has an arbitrary spin direction
Suppose we have to find the eigen values of spin operator $\hat{S}$ along a unit vector $\hat{n}$ that lies in XZ plane and then we are to find the probability measuring $S_z$ with $+\frac{\hbar}{2}$ ...
