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Questions tagged [eigenvalues]

Eigenvalues are a special set of scalars associated with a linear system of equations (i.e., a matrix equation) that are sometimes also known as characteristic roots, characteristic values (Hoffman and Kunze 1971), proper values, or latent roots (Marcus and Minc 1988, p. 144).

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What are some good medium matrices with known eigenvectors?

I am trying to test if an eigendecomposition I have is working properly. For this I would like some matrices that are 10x10 (ish) with trivial (or known) eigenvectors and eigenvalues so that I can ...
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Fastest way to calculate the eigenvector with the largest eigenvalue for a 3*3 positive-definite matrix [duplicate]

As stated in the title: I have a 3 by 3 positive-definite matrix $M$. What I need is the eigenvector corresponding to the largest eigenvalue, since I am calculating the solution to maximize the value ...
Enigmatisms's user avatar
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Eigenvalue Problem with Pseudospectral Chebyshev Polynomials

I am solving a linear 4th Order Eigenvalue ODE (Euler-Bernoulli Beam): $$ {\frac{d^{4}w}{dx^{4}}} = - \alpha {\frac{d^{2}w}{dx^{2}}} $$ The method I used was to apply a truncated spectral expansion ($...
Chlorine Pentoxide's user avatar
2 votes
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What is fastest method for finding the minimum and maximum eigenvalues of a (possibly very large) symmetric matrix?

What is the best way to find the extreme eigenvalues - in order to find the spectral radius - of a general real dense symmetric matrix? Looking at similar questions e.g.: What's the most efficient ...
ufghd34's user avatar
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2x2 complex symmetric eigendecomposition - LAPACK subroutine CLAESY

Asking here because I searched the LAPACK user forums and found nothing. I have a problem that requires the computation of the eigendecomposition $A=A^T=Q \Lambda Q^T$ for the 2x2 complex symmetric ...
coolguy1000000's user avatar
5 votes
2 answers
155 views

Optimized Lanczos method for finding eigenvalues of $A \otimes B$

Recently my supervisor told me about an efficient way to calculate eigenvalues and eigenvectors of matrix $A \otimes B$ with $a_{1} \times a_{2}$ as dimensions of $A$ and $b_{1} \times b_{2}$ is of $B$...
Mohammad. Reza. Moghtader's user avatar
3 votes
1 answer
209 views

Matrix Diagonalization and Computational Requirements

I have some questions about diagonalizing matrices. My interest lies in computing all eigenvalues of a given matrix. To avoid wasting time and improve my research efficiency, I want to understand the ...
yosuga's user avatar
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Parallel vs Serial computing using Python Qutip. Why is serial faster?

I am trying to learn parallel computing using qutip's parallel_map function. I've tried to write a basic and simple code to understand the differences between parallel and serial calculation of ...
muzbi's user avatar
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'eigs()' in Matlab gives inaccurate eigenvector when only several eigenvalues are calculated

I would like to report an issue which may be interesting in computational physics. Sometimes, to save time and memory, we use eigs() to calculate the first several ...
basic nutshell's user avatar
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Best transform of matrix to make it efficient for shift-then-invert?

I am using ARPACK to find the smallest eigenvalue of a matrix. I use the shift and invert method. That is, looking for the largest eigenvalue of $$ (A-\sigma I)^{-1}. $$ However, I do not know $\sigma$...
Ma Joad's user avatar
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Why for $A^T A$, it is faster to computer the eigenvalues of its inverse than itself?

I have written the following code in MATLAB. I also observe the same effect in Arnoldi iteration in ARPACK in C. ...
Ma Joad's user avatar
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Ways to speed up Lanczos algorithm when we have a very dense cluster of many eigenvalues?

Lanczos algorithm can be used to find the largest/smallest eigenvalues of matrices. I am trying to find a good library in C/C++/Rust for finding the smallest singular value (or eigenvalue). I have ...
Ma Joad's user avatar
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1 answer
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Eigenvalue problem and pseudoinverse of a product of sparse matrices

If I have some dense matrix that can be decomposed into a product of sparse matrices with known(but different) sparsity patterns. Can I somehow use this information to more efficiently compute its ...
HRI's user avatar
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using scipy.sparse.linalg.eigsh for degenerate states in Bose Hubbard model

I am currently writing a code for the Bose-Hubbard model, and I am calculating the ground states and single-particle density matrix for different values of U and J. As U=0, one would see how the ...
Lorenzo Carfora's user avatar
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Preconditioned GMRES for nearly diagonalizable systems

I have been working with a matrix $A$ and preconditioner $P\approx A^{-1}$ that I've then applied GMRES to the (left) preconditioned linear system \begin{equation} P^{-1}Ax=P^{-1}b \end{equation} $P^{-...
Tucker's user avatar
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13 votes
8 answers
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Real-world applications of eigendecomposition?

Cross-posted on Math.SE Are there real-world applications that call specifically for eigenvalues rather than singular values? I often see eigendecomposition used as "poor-man's SVD" For ...
Yaroslav Bulatov's user avatar
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eigenvalues of inhomogeneous Helmholtz equation violate superposition using FEM

I am trying to solve the in-homogeneous Helmholtz equation with damping and forcing using finite element method (FEM) with FEniCSx. The equation is; $c^2\nabla^2p - c^2\omega i d \nabla^2p + \omega^2 ...
Ekrem Ekici's user avatar
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How do you determine the Mott-Insulator to Superfluid transition in the Bose Hubbard System

I am doing some simulations on various systems expressed in 2nd quantization and one of the points of interest of mine was Phase transitions in the Bose-Hubbard model $$ H = \sum_{k} \{ t_k(b^\dagger_{...
Mephistopheles Faust's user avatar
12 votes
2 answers
2k views

Why are systems with clustered eigenvalues easy to solve?

I came across the following slide by Theo Diamandis & Zachary Frangella on what makes the linear system $Ax=b$ easy to solve using the conjugate gradient method. Transcription: CG converges ...
Yaroslav Bulatov's user avatar
3 votes
0 answers
135 views

Fast Fourier Transform on Meshes

I have a (closed, manifold, oriented) triangular mesh for which I build a matrix $L\in\mathbb{R}^{n\times n}$ discretising the negated Laplace-Beltrami operator. The matrix $L$ is symmetric positive ...
lightxbulb's user avatar
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1 vote
2 answers
148 views

Matlab eigs function with function handle

I am reading through the documentation of matlab function eigs specifically the function handle input version. Here it is: https://www.mathworks.com/help/matlab/ref/...
user8469759's user avatar
5 votes
1 answer
136 views

Numerical continuation of all eigenvalues of small, dense matrices

Consider a one-parameter family of matrices $A(q)\in \mathbb{R}^{n\times n}$, $q\in\mathbb{R}$. For my applications, $n$ is typically between $5$ and $50$ and $A(q)$ is generally dense, so direct ...
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Different computational approaches to show that the conjecture is true

It seems that proving the statement given in the question https://math.stackexchange.com/questions/4601532/a-pen-and-paper-proof-for-a-matrix-implication is difficult analytically. I was thinking what ...
BAYMAX's user avatar
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4 votes
1 answer
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Why are all eigen solvers iterative?

I have small dense square matrices for which I would like to compute the inverse by singular value decomposition, or equivalently solve the eigenvalue problem. While there are many direct methods for ...
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2 votes
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80 views

Diagonalization of large sparse matrix, computational programme recommendation and methods

According to this link, All eigenpairs of large sparse symmetric matrix. The guy @Baranas seems to have given a very confident answer about solving the whole Eigen spectrum. May I know if anyone has ...
Lee Zhi Yan's user avatar
1 vote
1 answer
185 views

Conditioning and Stability of generalized eigenvalue problem

The (generalized) eigenvalue problems with a multiple eigenvalue are the ill-posed ones. I have two questions that should be simple for experts: (1) Is the eigenvalue problem much more sensitive to ...
alpx's user avatar
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0 answers
91 views

Optimal Krylov subspace dimension and iteration limits for eigs

When using the eigs function in MATLAB, which is based off of ARPACK, one can manually modify the maximal dimension of the constructed Krylov subspaces, the maximum iteration counts, and the error ...
user45844's user avatar
3 votes
1 answer
391 views

Generalized eigenvalue problem for large, potentially ill-conditioned systems

Say that I have a generalized eigenvalue problem of the form $$Ax=\lambda Bx.$$ Using MATLAB, some naive ways that one may solve this is by either directly inverting $B$ then applying the ...
user45844's user avatar
1 vote
2 answers
796 views

How should I solve generalized eigenvalue problems in Python? (Orr-Sommerfeld equation)

I am trying to solve the Orr-Sommerfeld equation numerically, using the techniques given in this article. This leads to solving a generalized eigenvalue problem, that is, given two matrices $\mathbf A,...
K.defaoite's user avatar
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0 answers
80 views

Rayleigh-Ritz under LOBPCG : Nested iteration?

I am trying to understand the basic LOBPCG algorithm, as used in popular python libraries, primarily from the suggested resource on wiki. LOBPCG involves a subroutine to compute the coefficients of ...
Shark's user avatar
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3 votes
0 answers
72 views

Approximating eigenvalues of DPR1 matrix with special properties

In my application, I have a sum of diagonal $A$ and rank-1 $B$ $$T=\underbrace{\text{diag}(1-2\alpha h+2\alpha^2 h^2)}_A + \underbrace{\alpha^2 hh^T}_B$$ Where $h$ is a vector $\in \mathbb{R}^d$ with ...
Yaroslav Bulatov's user avatar
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0 answers
58 views

Eigenvalues of same operator expressed in two different orthonormal basis are coming out different

I have an operator $H$. I express $H$ as a matrix in the orthonormalized $\{ |e > \}$ basis. Then I diagonalize it to obtain eigenvalues, let's say for example $H$ is $6 \times 6$ and the ...
Snpr_Physics's user avatar
5 votes
1 answer
177 views

Eigenvalues of diagonal plus rank-one

I need to compute an eigendecomposition of an $n\times n$ matrix $$ D + c vv^\top = Q\Lambda Q^\top \tag{1} $$ in MATLAB, where $D$ is a real diagonal matrix, $c$ is a scalar, and $v$ is a real vector....
eepperly16's user avatar
2 votes
1 answer
146 views

qr-algorithm to find eigenvalues not returning expected values

I tried to compute eigen values with the QR-algorithm found here (there is also a wikipedia page also) ...
roi_saumon's user avatar
0 votes
2 answers
480 views

Solving a generalized eigenvalue problem with Chebyshev spectral method: How to impose boundary conditions into the matrices?

I'm solving a local instability problem for a pipe Poiseuille flow. The coordinate system is columnar, i.e., ($r,\theta,x$) (radial, tangential and axial). The basic flow is $\bar{u_r}=0, \bar{u_\...
Jack's user avatar
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2 votes
0 answers
45 views

Iterative methods for underestimate of smallest eigenvalue for large sparse matrices

I recently read the paper "EUCLIDEAN-NORM ERROR BOUNDS FOR SYMMLQ AND CG" by Estrin et al. and there they use an underestimate (i.e. something in $(0,\lambda_{min}]$) of the smallest ...
lightxbulb's user avatar
  • 2,257
2 votes
0 answers
205 views

Generalized Eigenvalue Problem using MATLAB

I'm trying to solve a generalized eigenvalue problem. I have two matrices $H$ and $S$ such that: $$ HX=λSX $$ I need to find the eigenvalues $\lambda$. The matrices $H$ and $S$ are real, asymmetric, ...
Beginner Noob's user avatar
2 votes
1 answer
123 views

Finding the correct order of eigenvectors of a parameter-dependent Hermitian matrix

so, I have a symmetric, analytic matrix $\mathbf{H}(x)$ ($x$ is real). Because $\mathbf{H}(x)$ is analytic and $x$ is real, it is possible to find analytic functions for the eigenvectors and the ...
cheetah's user avatar
  • 153
2 votes
1 answer
152 views

Numerical diagonalization of Hamiltonian

Framework I am trying to diagonalize the Bogoliubov-de Gennes Hamiltonian. The problem is that the Hamiltonian contains a Laplacian. This could be solved by using a discretized Laplacian. How I tried ...
Guest's user avatar
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1 vote
0 answers
113 views

Well-conditioned pseudospectral for computing eigenvalues to (partial) differential equations

I am working on writing a Chebyshev pseudospectral method (see for example "Chebsyhev and Fourier spectral methods" by John Boyd) to solve for the eigenvalues of differential equations of ...
physics_researcher's user avatar
1 vote
1 answer
1k views

How to implement Lax-Friedrich flux splitting with WENO scheme

I'm working on modeling a shock wave using the Euler equation with an advanced Equation of state and the fifth order WENO scheme. The equation are set up on the form: \begin{equation} \frac{\partial U}...
Twm1995's user avatar
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2 votes
1 answer
194 views

LOBPCG bad preconditioned performance for largest eigenpairs

The LOBPCG algorithm finds eigenpairs of the generalized eigenproblem $$ Ax = \lambda B x $$ where $B$ is symmetric and positive-definite, $A$ is symmetric. One of the features that makes LOBPCG so ...
Nico Schlömer's user avatar
0 votes
0 answers
76 views

Eigen-decomposition one eigenpair by one eigenpair?

Is it possible to conduct an Eigen-decomposition of a matrix one eigenpair by one eigenpair? And related to this question, what is the time complexity of truncated eigendecomposition? I am trying (...
Tan's user avatar
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7 votes
0 answers
142 views

Can we sparse solve a few eigenvalues specified by index range?

I need to solve a few eigenvalues of a large sparse matrix specified by their index range. These indices are according to the whole eigenspectrum sorted in algebraic (not absolute value) ascending ...
xiaohuamao's user avatar
1 vote
1 answer
1k views

Numerical Solution of the Schrödinger equation for hydrogen

I'm trying to solve the Schrödinger equation for the hydrogen atom in the following form numerically: $$\left[-\frac{\hbar^2}{2m}\frac{d^2}{dr^2}+V(r)+\frac{\hbar^2l(l+1)}{2mr^2}\right]R(r)=ER(r).$$ ...
mysterion123's user avatar
3 votes
1 answer
563 views

A method for finding the number of eigenvectors with a given, known eigenvalue

Is there a method for finding the number of eigenvectors with a given eigenvalue? I do not need the eigenvectors themselves, and to find the eigenvectors seems quite tough, given the comments on the ...
user196574's user avatar
4 votes
1 answer
103 views

Roundoff errors in FEM computations - generalized eigenvalues

This is a continuation of my previous question. I am trying to effectively compute a bound for the roundoff errors in some FEM computation (2d polygons, triangular meshes). Below I will write some of ...
Beni Bogosel's user avatar
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1 vote
1 answer
188 views

Accelerated Inverse Power Method with Rayleigh Quotient

I am considering implementing the accelerated inverse power (AIP) method with Rayleigh quotient to speed up eigendecomposition of a real square symmetric matrix. Halton (1996) gives an example ...
user avatar
3 votes
4 answers
591 views

Solving the eigenvalue from a set of coupled second order differential equation numerically

I met a problem in solving a set of coupled differential equation, as shown below: $$A_1\psi_1(z)+A_2\frac{d^2\psi_1(z)}{dz^2}+A_3\frac{d\psi_2(z)}{dz}=\lambda\psi_1(z)$$ $$A_4\psi_2(z)+A_5\frac{d^2\...
JensenPang's user avatar
19 votes
3 answers
3k views

Why do we usually not want the eigenvalues of non-symmetric matrices?

I came across this line in a class note I am reading where it discusses finding eigenvalues of matrices. In reality we don't go all the way with Arnoldi. We stop at a decent value of 𝑘. Then the 𝑘 ...
CuriousMind's user avatar

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