# 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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### Linear stability in gravity driven Flow numerical solution

I am trying to solve a gravity driven flow problem for thin films and I am having some difficulties solving the PDE resulting from the linear stability analysis for the steady state solution. This has ...
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### 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 ...
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### 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, ...
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### 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 ...
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### Oscillating eigenvectors for 2d-laplace operator

I am trying to calculate the eigenvectors of a Laplace-operator in 2d, with boundary conditions equal to $u=0$ if $x, y$ are outside of a rectangle defined as $(0.5, 0.5), (1.5, 1.5)$. For that I used ...
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### 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 ...
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### 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 ...
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### 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}...
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### 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 ...
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### 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 (...
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### 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 ...
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### How to obtain smallest eigenvalues with Arnoldi iteration

I understand that the Arnoldi iteration produces a basis which tends to include in its span the eigenvectors corresponding to eigenvalues of large magnitude (hence the analogy between the last vector ...
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### Numerical solution to the infinite well problem

I've used the following code to implement it ...
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### Reproducing a paper's result for Topological Insulators

For the past weeks I have been trying to reproduce Agarwala's results but I've been unsuccessful. From this paper I am trying to reproduce the first and last columns of Fig.2, by implementing eq.2; ...
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### eigsh (Lanczos algorithm) slows down for degenerate eigenvalues

I have a complex Hermitian matrix of size about $70000\times 70000$. I want about 100 eigenvalues near 0. However, I know that every eigenvalues are two-fold degenerate. I found out that the running ...
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I am trying to compute the natural frequency of a cantilevered beam. The Euler-Bernoulli equation reduces to the following problem : $$v''''=\lambda v, \text{with }, v(0)=0, v'(0)=0, v'''(1)=0,v''(1)... 0 votes 1 answer 81 views ### How are eigenvalues of a cell in the finite volume discretization calculated? (for euler fluid equations) I am building a program to numerically solve the euler fluid equations based on finite volume discretization and am having trouble on a step where the eigenvalues of a cell need to be calculated ... 3 votes 0 answers 72 views ### Does shift-invert method has invertibility issue? Please note that I have nearly zero background on numerical methods. I understand the shift invert method as described in SciPy Tutorial The main argument of the above link is as follows. Suppose we ... 2 votes 4 answers 283 views ### Eigenvalue decomposition for a very huge matrix of medical images (such as the pixel physical coordinates of CT images) I am trying to do eigenvalue decomposition for a huge matrix larger than 788000×788000 for medical image analysis. The matrix is not sparse and every element in the matrix has a real value. And, for ... 0 votes 1 answer 223 views ### Pseudospectrum of non square Matrix in Python I have a rectangular matrix A \in \mathbb{R}^{m \times n} ... 3 votes 1 answer 155 views ### Lanczos algorithm for finding top eigenvalues of a matrix sum I am trying to find the top k leading eigenvalues of a NumPy matrix (using python dot product notation) L@L + a*X@X.T, where L ... 0 votes 0 answers 293 views ### Numerical Range of a matrix in Python In the mathematical field of linear algebra and convex analysis, the numerical range or field of values of a complex n\times n matrix A is the set$$W(A)=\left\{{\frac {{\mathbf {x}}^{*}A{\... 100 views

### Large scale nonsymmetric eigenvalue problems in practical applications

I am trying to determine need for solving large scale nonsymmetric eigenvalue problems in both industry and academia. I am interested in any kind of problem where we cannot assume that matrices are ...
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### Given a symmetric matrix, is it ok to apply Cholesky decomposition to see if it has negative eigenvalues?

I intend to check the diagonal of L, where A = L'L, for negative elements. However, I don't know if Cholesky is meaningful in theoretical / computational sense if there are some negative eigenvalues.
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### Neural networks trained with subspace iteration algorithms for solving SCF eigenvalue problems

I am toying with the idea of using current libraries available with SCF (Self-Consistent Field) subspace iteration codes to train an ANN (Artificial Neural Network) to solve for SCF eigenvalue ...
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### Accuracy issues with Arpack in Julia for eigenvalues of smallest magnitude

Following the documentation of Julia's Arpack package (Cf. https://julialinearalgebra.github.io/Arpack.jl/stable/eigs/) I have computed some largest and smallest magnitude eigenvalues of sparse ...
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### Quick way to find a common basis of eigenvectors between 2 matrices : valid or not?

Following the advise of @Federico Polonion a previous post, one suggested, to find a basis of common eigen vectors between 2 matrices, to simply do : Generate 2 ...
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### Faster eigenvector routine for non-symmetric matrices with real eigensystem?

I have non-symmetric real-valued matrices with real-valued eigensystems. How to compute eigenvectors efficiently? Using scipy.linalg.eig (which calls ...
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### Can this simple quadratic optimization problem be turned into a simple eigenvalue problem?

I'm interested in a type of problem on this form $$\min_{x} x^{T}Ax+x^{T}b \quad \text{s.t} \quad x^{T}x=1$$ where $A$ is positive definite. As you can see, if it weren't for the $x^{T}b$ term in the ...
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### Cheap recalculation of eigenvalues and eigenvectors for a low-rank update of the matrix

Suppose I have a correlation matrix, $A$, and I already have the eigenvalues and eigenvectors of this matrix. For a given vector, $\mathbf{\mathit{v}}$, I want to calculate the eigenvalues and ...
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### Are there any constraints on eigenvalues that are used in inverse iteration?

What is the result of the method for multiple eigenvalues? Is there any case for which this method will not work altogether?
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### Numerical scheme to calculate the normal mode of a set of hyperbolic PDEs?

I would like to solve the linearised, ideal, MHD equations, where the gas pressure is zero. \frac{\partial u_x}{\partial t}=v_A^2(x,z)\left[\nabla_{||}b_x - \frac{\partial b_{||}}{\partial x}\right],...
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### Why is LAPACK (seemingly) suboptimal for packed and banded eigenvalue problems?

Based on this LAPACK routines list, it looks like there is no relatively robust representation (RRR) driver routine for either packed or banded symmetric eigenvalue problems. According to the relevant ...
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### Is LAPACK behind the cutting edge of dense linear algebra?

I have been digging into some numerical linear algebra lately, and reading in particular about how LAPACK solves symmetric eigenvalue problems. I noticed that the ...
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