All Questions
Tagged with linear-algebra mp.mathematical-physics
49 questions
-5
votes
1
answer
86
views
Why is the second order correction to energy zero for a fully degenerate eigensystem? [closed]
Consider the system given by,
$$ H|n\rangle = E|n\rangle$$
where:
$H$ is the hamiltonian.
$|n\rangle$ is the eigenstate.
$E$ is the energy of the eigenstate.
Using degenerate perturbation theory and ...
- 23
5
votes
1
answer
539
views
Under what circumstances Is a symmetric matrix representable as a Coulomb matrix?
Question:
I am exploring a neural network architecture inspired by physical interactions, where each neuron has associated "mass" and "position" vectors. The weight matrix between ...
- 3,064
2
votes
0
answers
60
views
Basis vectors using anti-commuting operators?
Let $V$ be a finite-dimensional inner product space. Suppose $A_{1},...,A_{N}$ are anti-commuting operators, meaning that these are linear operators on $V$ that satisfy:
$$A_{i}A_{j}+A_{j}A_{i} = A_{i}...
- 1,305
0
votes
0
answers
121
views
Representation of anti-commuting matrices in $\mathbb{C}^{2}$
This is a cross posting updated question from MSE. I have not got any answers there yet and I really want to understand this problem.
The basic question is the following. Let $V$ be a finite-...
- 1,305
5
votes
0
answers
202
views
Difficulty of solving $Ax=b$ in terms of limiting spectral density of $A$?
Suppose $A$ is a random real-valued $n\times n$ matrix and we want to know the difficulty of solving $Ax=b$ when entries of $b$ are sampled IID from Normal$(0,1)$.
Can we say anything about the ...
- 2,242
5
votes
1
answer
167
views
What structure is preserved by pseudo-homeomorphisms of pseudo-Euclidean spaces?
Let us recall that for integer numbers $t,s\ge 0$ the pseudo-Euclidean space $\mathbb R^{t,s}$ is the vector space $\mathbb R^{t+s}$ endowed with the quadratic form $q_{t,s}:\mathbb R^{t+s}\to\mathbb ...
- 41.9k
2
votes
1
answer
456
views
On a theorem of Carlson on the necessary and sufficient condition for a matrix to have $m$ real eigenvalues
Background: In the physics of open quantum systems the Lindbladian $\mathcal{L}$ governs the evolution of quantum states through the Lindblad master equation.
The Lindblad operator usually has ...
- 1,054
2
votes
2
answers
243
views
Sum over exponentiated bilinear form in finite-field vector space
Let $A$ be a linear map over the finite-field vector space $(\mathbb F_2)^n$, i.e., an $\mathbb F_2$-valued $n\times n$ matrix, not necessarily symmetric. I'm interested in the sum
$$Z(A) = \sum_{X\...
- 3,001
1
vote
0
answers
46
views
Find $\max_V \text{Tr} \left((\rho_2 (V \otimes I) \rho_1 (V^\dagger \otimes I)\right)$
I am doing a quantum optimization where the final problem has the following form
$$\max_V \text{Tr} \left((\rho_2 (V \otimes I) \rho_1 (V^\dagger \otimes I)\right),$$
where $V \in \mathbb{C}^{d\times ...
- 11
4
votes
3
answers
321
views
Vacuum vector and basis defined by anti-commuting operators
Let $\mathcal{H}$ be a finite-dimensional inner product space over $\mathbb{C}$. Suppose $A_{1},...,A_{N}$ are linear operators on $\mathcal{H}$ such that:
$$\{A_{i},A_{j}\} = 0 \quad \mbox{and} \quad ...
- 1,305
-3
votes
1
answer
134
views
SU(2) and entangled particles [closed]
We have two particles $A$ and $B$ in a maximally entangled state $|\Psi\rangle \in \cal{H}_A \times \cal{H}_B$
$$
\left|\Psi\right\rangle = \frac{1}{\sqrt{2}} ( \left| 0
\right\rangle_A\otimes \left| ...
- 1
10
votes
1
answer
337
views
What are the predictive implications of conditional non-commutative probability?
To simplify things, let's consider the Hilbert approach to quantum probability over a finite dimensional vector space $V$ of dimension $n$.
In this context a state $S$ is a positive semi-definite ...
- 323
0
votes
0
answers
69
views
Formulas involving traces of products of singular hermitian positive semidefinite $2$ by $2$ matrices
While working on the Atiyah problem on configurations of points, I came across formulas involving products of traces of products of singular hermitian positive semidefinite $2$ by $2$ matrices.
To ...
- 5,215
3
votes
0
answers
2k
views
Multiplication of two Pauli string
Given a Pauli string $P_i \in \{ I,X,Y,Z\}^{\otimes n} $
Example: $P_0 = XXYIZ = X \otimes X \otimes Y \otimes I \otimes Z $.
Here $I,X,Y,Z$ are Pauli matrices defined explicitly as:
$$
I = \begin{...
- 131
13
votes
1
answer
592
views
Is there a straightforward generalization of min(x,y) to positive-semidefinite Hermitian matrices?
This is an open-ended question I have. Is there a function of two positive-semidefinite hermitian operators $\min(A,B)$ returning another positive-semidefinite Hermitian operator such that:
If A and ...
- 1,947
2
votes
0
answers
87
views
Entropy per site of quantum spin chain
It’s fork lore that von Neumann entropy (and free energy) grows linearly with respect to the size of a quantum system. Is there a rigorous demonstration in the toy model of a quantum spin chain with (...
- 189
5
votes
2
answers
296
views
On the “Non-conservation of parity in weak interactions”
Kostrikin and Manin, in their Linear Algebra and Geometry, state that:
(The excerpt is on pp. 42-43.)
The statement comes after a proof of general linear group over reals having two connected ...
- 258
1
vote
1
answer
247
views
Linearity of the directional derivative of a convex functional at the minimum
Let $H$ be a Hilbert space, $T_+(H)$ the set of positive self-adjoint trace-class operators on $H$, and $f : T_+(H) \to [0,m]$ a non-negative, bounded, convex functional. I don't necessarily know that ...
- 695
3
votes
1
answer
151
views
Commutation between integrating and taking the minimal eigenvalue
Let $S = (f_{ij})_{ij}$ be a $n \times n$ real symmetric matrix, with functions $f_{ij} \in L^1(\mathbb{R}^d,\mathbb{R})$ in it. We define $\left(\int u S \right)_{ij} = \int u S_{ij}$ as the ...
user140442
1
vote
1
answer
166
views
Understanding the Time Delay of Arrival trilateration algorithm
I'm trying to algorithmically solve the Time Delay of Arrival problem as part of some mathematics research. The problem is as follows:
Given the location of three receivers in a plane (A, B, and C), ...
- 111
11
votes
2
answers
353
views
Exponential decay of voltage potential difference
Consider the following adjacency matrix of a complete graph:
$$A=(e^{-|i-j|})_{1\leq i\neq j\leq n}$$
with 0 on the diagonal. Let $D=diag\{d_1,...,d_n\}$ be the degree matrix where $d_i=\sum_{j\neq i}...
- 1,495
4
votes
0
answers
457
views
Inequalities for trace/eigenvalues of product of multiple 2x2 matrices
Consider the matrix product $\prod_i^n A_i$,
where each $A_i$ is a $2\times2$ matrix having the form $A_i = \left( \begin{smallmatrix} \lambda + \alpha_i & -\beta_i \\ 1 & 0\end{smallmatrix}\...
- 695
6
votes
1
answer
321
views
Branching from $E(6)$ to $SO(10) \times U(1)$
In $E(6)$ inspired models of supersymmetry, the inclusion of Lie subgroups
$$
SO(10) \times U(1) \hookrightarrow E_6
$$
is important object of interest. See here for my motivating example.
In ...
- 835
1
vote
0
answers
103
views
Which operators constructed from 10d gamma matrices commute with $SO(1,2)\times SO(3)\times SO(3)$?
In the paper Supersymmetric Boundary Conditions in N=4 Super Yang-Mills Theory by Gaiotto and Witten, an in-depth analysis of boundary conditions in N=4 Super Yang-Mills in four dimensions in ...
- 1,155
14
votes
1
answer
738
views
For a stable matrix $B$ and anti-symmetric $T$, such that $B(I+T)$ is symmetric, show that $\mbox{tr}(TB)\leq0$
Let stable matrix (i.e., its eigenvalues have negative real parts) $B \in \mathbb R^{n \times n}$ and anti-symmetric matrix $T \in \mathbb R^{n \times n}$ satisfy
$$B^\top - T B^\top = B + B T$$
...
- 371
4
votes
2
answers
2k
views
Advanced reference and roadmap about random matrices theory
There is few posts on MO that asked about reference on this topic, and I found some difficulty during the process of getting myself into the subject so here is the question.
I really want to hear ...
11
votes
2
answers
714
views
A neat evaluation of an infinite matrix?
Let $M_n$ be an $n\times n$ matrix defined as
$$M_n
=\left[\frac{2i+1}{2(i+j+1)}\binom{i-1/2}i\binom{j-1/2}jx^{i+j+1}\right]_{i,j=0}^n.$$
With $I_n$ the identity matrix, consider $A_n:=I_n-M_n^2$. ...
- 43.2k
5
votes
2
answers
389
views
Pfaffian of several skew-linear transformations / matrices
Introduction: Let's assume we have a 2-form $\alpha=(1/2)\sum_{j,k=1}^n a_{jk}\ e_j\wedge e_k$, where $n=2m$, and $a_{jk}\in\mathbb C$. We know that $\alpha^{\wedge m}=\alpha\wedge\alpha\dots\wedge\...
- 53
11
votes
2
answers
964
views
How to prove this determinant is positive-II?
Question: Given an arbitrary number of real matrices of the form $ A_i=
\biggl(\begin{matrix}
C_i+E_i & B_i \\
B_i^T & D_i-F_i
\end{matrix} \biggr)
$, where $B_i$ is an arbitrary $n\times n$ ...
- 845
8
votes
1
answer
558
views
What is the spin connection in 9 dimensions as opposed to 5 dimensions?
From Spin Connection in 5 dimensions I can define a massless fermion's covariant derivative on a curved manifold as
$$
\nabla_\mu \psi = (\partial_\mu - {i \over 4} \omega_\mu^{ab} \sigma_{ab}) \psi
\...
- 215
9
votes
0
answers
360
views
Finding $U,V$ in Thompson's Formula
Thompson's formula says, given $A,B \in \mathfrak{su}(n)$, there exists $U,V \in SU(n)$ such that:
$e^{A}e^{B}=e^{UAU^{\dagger} + VBV^{\dagger}}$
Given $a,b \in \mathfrak{su}(4)$ defined by:
$a=J_x ...
- 2,099
3
votes
1
answer
212
views
Possible lower bound in quantum many body system with non-local terms
I am asking a question related to Lieb-Robinson bound and nonlocality.
As we know from Lieb-Robinson theorem (see e.g. http://arxiv.org/abs/1008.5137): Suppose a Hamiltonian system is local, i.e. $H=\...
- 91
0
votes
1
answer
130
views
Reference for measures of commutativity needed
I'm looking for an appropriate measure to quantify the extent to which two matrices commute. In other words, if A and B are two n×n Hermitian matrices, and [A,B]=C.
I'd like a function μ:Cn×n→[0,∞) ...
- 209
7
votes
2
answers
992
views
Is there a nice "synthetic" way for doing differential geometry on infinite dimensional vector spaces?
If $V$ is an infinite dimensional vector space, for example the space of smooth functions on $\mathbb{R}$, we can introduce some differential geometry concepts by choosing a topology on $V$ and doing ...
- 922
6
votes
1
answer
353
views
Quaternion Wishart matrices of half-integer dimension?
For a physics application (quantum delay times of a chaotic scatterer) I need to generate $m$ positive random variables $\lambda_1,\lambda_2,\ldots\lambda_m$ with probability distribution
$$P_\beta(\...
- 188k
1
vote
2
answers
747
views
Can any antilinear involution be trivialized by a change of basis?
Consider an antilinear involution, that is an antilinear map on a complex vector space, whose matrix $M$ obeys $MM^*=1$ where the star denotes complex conjugation. Can we find a change of basis whose ...
- 378
-3
votes
1
answer
232
views
A problem that involves matrix and Lorentz Transformation [closed]
To be clear I address the question in two parts as below. All matrixes involved are real four-dimensional matrixes.
$1.$Let $G$ be the matrix $diag(1,-1,-1,-1)$. $A$ is a matrix satisfying $A G A^T=A^...
- 45
1
vote
0
answers
214
views
range of the difference-of-two-qubit-$4 \times 4$-density-matrix-determinants
The determinant of a two-qubit $4 \times 4$ density matrix--that is, a Hermitian, nonnegative definite matrix with unit trace--lies between $0$ and $(\frac{1}{2})^8$. (A "pure state" has determinant ...
- 723
0
votes
0
answers
257
views
What is the integer form of a projector into the intersection of the ranges of two integer projection matrices?
Consider two square integer matrices $X$ and $Y$ of the same dimension with the following properties:
$X^2=rX$, and $Y^2=sY$ for integers $r$ and $s$. The $\gcd$ of the entries of $X$ is 1 and the $\...
36
votes
0
answers
2k
views
Correspondence between eigenvalue distributions of random unitary and random orthogonal matrices
In the course of a physics problem (arXiv:1206.6687), I stumbled on a curious correspondence between the eigenvalue distributions of the matrix product $U\bar{U}$, with $U$ a random unitary matrix and ...
- 188k
9
votes
1
answer
611
views
Matrix-tree theorem via supersymmetry (i.e. Grassman algebras)
The matrix-tree theorem states the number of spanning trees of a graph $G$ is equal to a modified determinant of the adjacency matrix or "graph Laplacian", $\Delta_G$:
$$\#\{ \text{spanning ...
- 22.8k
8
votes
2
answers
583
views
Efficiently computing a few localized eigenvectors
Let $H = \triangle + V(x) : \mathbb{R}^2 \rightarrow \mathbb{R}^2$. I am interested in domain decomposition for an eigenproblem involving $H$.
The lowest 1000 eigenfunctions of $H$, $ \psi_i $, can ...
- 817
32
votes
3
answers
3k
views
How much linear algebra can be done with graphs?
Let G be a finite directed acyclic graph, with sources $A=\{a_1,\ldots,a_n\}$ and sinks $B=\{b_1,\ldots,b_n\}$, with edge weights $w_{ij}$. The weight of a directed path P is the product of weights of ...
- 22.1k
9
votes
4
answers
2k
views
Is there a useful generalization of the Schmidt decomposition to the tensoring together of 3 or more vector spaces?
I've rewritten the question in math notation, and I've left the old version in physics bra-ket notation here.
Background
A simple consequence of the singular value decomposition is that any vector $...
- 846
2
votes
0
answers
1k
views
Can we pass to the limit in Poincaré-Jaynes-Bretthorst interpolation and deconvolution?
In Science and Hypothesis, chapter XI, The calculus of probabilities, Henri Poincaré deals informally with the fundamental problem of interpolation. He concludes (see http://ia600308.us.archive.org/21/...
3
votes
1
answer
572
views
When is a finite matrix a "good" approximate representation of an operator?
I am interested in representing an arbitrary charge density (say, of atoms in a molecule) $\rho(r), \; r\in \mathbb{R}^3$ by a finite linear combination of basis functions
$\rho(r) = \sum_{i=1}^N q_i ...
- 1,890
4
votes
2
answers
1k
views
Under what conditions will a unitary matrix fix a subspace which does not diagonalize the generating Hamiltonian?
Hello, this is my first post here. I hope that it is not too vague; I will be as precise as I can, but I have more of a meta-problem so please forgive me if this is inappropriate. My question is ...
5
votes
2
answers
687
views
Dependence of trace norm on matrix size for smooth vs. random matrices.
Problem
Consider two d x d complex matrices, R and S, whose entries lie in the unit disk:
$\quad |R_{i,j}|<1 \quad$ and $\quad |S_{i,j}|<1 $.
Say that R is constructed by randomly choosing ...
- 846
3
votes
1
answer
632
views
What is the entropy of a density matrix which is the sum of two unitarily equivalent projectors?
Construction
Suppose I have a density matrix $\rho$ which is proportional to a projector $P$ formed by tensoring together $N$ small projectors $P^{(i)}$ of rank 2:
$P^{(i)} = |a\rangle_i\langle a| + |...
- 846