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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53 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 ...
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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 ...
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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 ...
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148 views

Pseudospectrum of non square Matrix in Python

I have a rectangular matrix $A \in \mathbb{R}^{m \times n}$ ...
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82 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 $L$ ...
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88 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{\...
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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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111 views

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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217 views

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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122 views

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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85 views

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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A misunderstanding or a bug in LAPACK's solver for generalized eigenvalue problems?

In my application, I have two general real matrices $A$,$B$ defined as follows, $$ A=\begin{bmatrix} -s I_3 & A_0 & 0 & 0 \\ A_0^T & -s I_3 & 0 & 0 \\ 0 &...
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Python scipy eigh(Arpack) giving wrong eigenvalues for generalized eigenvalue problem

I am trying to solve a generalized eigenvalue problem using Arpack, right now the code is using LAPACK but that's too slow, we only need a few eigenvalues and the matrices are sparse so using Arpack ...
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1answer
86 views

Jacobi iterative method

I'm using Jacobi iterative method for finding eigenvalue and eigenvector for hermitian or symmetric matrix. Eigenvectors corresponding to eigenvalues are not exact. The third eigenvector is totally ...
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1answer
99 views

Efficient solution to a structured symmetric linear system with condition number estimation

I have a real-valued linear system $Hx = b$ where $H$ is symmetric matrix** (not necessarily positive/negative definite) with a very particular structure: $$ H = \begin{bmatrix} D && B \\ B^T &...
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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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687 views

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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1answer
174 views

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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1answer
55 views

Newman algorithm yielding different result to what is given in his paper

Summary I am trying to implement Newman's algorithm for community detection, outlined in this paper. I am testing my implementation against one of the datasets used in that paper to benchmark the ...
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1answer
445 views

How can I evaluate more accurate energy eigenvalues from Schrodinger equation using shooting method?

I am trying to use the "shooting method" for solving Schrodinger's equation for a reasonably arbitrary potential in 1D. But the eigenvalues so evaluated in the case of potentials that do not have hard ...
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166 views

Implementation of a $O(n \log(n))$ method to compute eigenvalues of real symmetric tridiagonal matrices

I just came upon this paper, which details the implementation of a fast method to get eigenvalues of tridiagonal symmetric matrices : Coakley, Ed S.; Rokhlin, Vladimir, A fast divide-and-conquer ...
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Efficient way to find eigenvalues of complex symmetric matrix with real off-diagonal elements

My goal is to find all eigenvalues (and eigenvectors) in a given range of magnitudes of a complex symmetric matrix with real off-diagonal elements (only diagonal elements are complex). Currently I'm ...
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2answers
158 views

2d Schrodinger Equation via matrix diagonalization in C

I am trying to solve the time-independent Schrodinger equation in two dimensions via discrete matrix diagonalization. I want the energy eigenvalues and the corresponding eigenfunctions for a given ...
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When does reduced integration lead to artificial zero energy modes in stiffness matrix?

This question relates to the topic of locking free finite element development. In the case of application of reduced integration to global stiffness matrix for the Timoshenko beam element with ...
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1answer
369 views

How to know which LAPACK's function is used by Scipy's eig function?

As far as I understood, scipy.linalg.eig use wrappers from scipy.lapack to compute the eigenvalues and eigenvectors of a matrix. ...
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1answer
290 views

Fastest way to calculate the $2$-norm (or an upper bound for the $2$-norm) of the inverse of a matrix $A\in \mathbb{C}^{N\times N}$

I have a matrix $A\in \mathbb{C}^{N\times N}$ and I need to calculate $||A^{-1}||_{2}$ efficiently. Can it be done without having to evaluate the inverse explicitly? In general, I am looking for ...
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2answers
595 views

Implementing Gelfand’s formula for the spectral radius in Python - lack of convergence

For context: Gelfand's formula for the spectral radius is $\lim_{k\rightarrow \infty}|A^k|^{1/k}$ where $|\cdot|$ is any well-defined operator norm. I naively coded a function to calculate the $k$th ...
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58 views

Fast algorithm for computing lower mode shapes and natural frequencies in MATLAB using sparse stiffness and mass matrices

I am looking for a fast algorithm for computing eigenvalues and eigenvectors from sparse stiffness and mass matrices in MATLAB. The eig(K, M) doesn't work with ...
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1answer
166 views

Reference for QR algorithm for complex matrix

I am trying to find out if the known QR algorithm to find the eigenvalues of a real matrix, which can be found in the book Fundamentals of Matrix Computations, can also be used for complex matrices ...
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285 views

How does gmres method iteration behave for this non-diagonalizable matrix?

Recently, I have been studied my lessons about gmres iteration, probably the most popular iteration method for general large sparse linear system of equations Ax=b. And the convergence is obtained ...
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1answer
261 views

How to compute all the eigenvalues of a large sparse matrix using matlab?

In matlab, there are 2 commands named "eig" for full matrices and "eigs" for sparse matrices to compute eigenvalues of a matrix. And eig(A) computes all the eigenvalues of a full matrix and eigs(A) ...
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1answer
219 views

How to perform an eigendecomposition of a general complex matrix with arbitrary precision in C/C++

I need to obtain the Eigenvectors of a general complex matrix, but with quadruple precision. Is anyone aware of a means to do this? I currently use Tux Eigen, and I see that in their unsupported ...
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1answer
64 views

Analytic formula for $\arg\max_{\|z\|_\infty \le 1}z^T A z$, where $A=uu^T+vv^T$

Let $u$ and $v$ be column vectors of size $n \gg 1$ (not both zero), and consider the matrix $A:=uu^T+vv^T$ Question What is an analytic formula for $\arg\max_{\|z\|_\infty \le 1}z^TAz=\arg\max_{\|z\...
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1answer
127 views

Analytic formula for leading eigenvector of $uu^T + vv^T$?

Let $u$ and $v$ be nonzero column vectors of size $n$ and consider the $n \times n$ positive-definite matrix $A:=uu^T + vv^T$. In this post https://math.stackexchange.com/a/112201/168758, the ...
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1answer
99 views

Efficient computation of leading eigenvector of a matrix product of the form $ADA^T$, where $D$ is diagonal

Let $A=[A_1|\ldots|A_m] \in \mathbb R^{n \times m}$ with $n \gg m \gg 1$ and $D=\text{diag}(d_1,\ldots,d_m)$ where $d_1,\ldots,d_m > 0$, and consider the $n\times n$ positive-definite matrix $X=\...
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Eigenfaces Algorithm

This might be a silly quesntion but recently I've been trying to program the eigenface algorithm using PCA, so I arranged the face vectors vertically in a matrix X such as: X = [x1,x2,x3,...,xn]; In ...
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2answers
627 views

Fast and accurate eigenvalue computation for 3x3 posdef matrices

I am looking for a very fast and efficient algorithm for the computation of the eigenvalues of a $3\times 3$ symmetric positive definite matrix. The algorithm will be part of a massive computational ...
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2answers
148 views

Singular values of $X$ in $AX+XA=C$?

Suppose I have semi-positive definite matrices $A$ and $C$, is there an efficient approach to get top singular values of X entering the following expression? $$ AX+XA=C $$ My matrices are 4k-by-4k ...
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1answer
157 views

An optimization method for bounding the eigenvalues of a unknown non symmetric matrix

Given a positive objective function $f$ that acts on a real-valued matrix $A$, I am interested in the following problem $$\underset{A \in \mathbb{R}^{n \times n}}{\text{minimize}} \quad f(A) \quad \...
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1answer
198 views

Calculate amount of FLOPs for an eigenvalue problem solver

I have 2 complex, non-symmetric, matrices $A_{1000\times1000}$, $B_{1000\times1000}$ and I am using Matlab to get it's eigenvalues (functions like eig or eigs). Both matrices are different - one is ...
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Simulating Anderson model, have problem with momentum representation (MATLAB)

I want to change from real-space representation to momentum-space representation I have a Hamilton-operator (Anderson-model), and I calculated some kind of entropy of its eigenstates (this is working, ...
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2answers
418 views

Convexity of Sum of $k$-smallest Eigenvalue

If I have a real positive definite matrix $A\in\mathbb{R}^{n\times n}$, and denote its eigenvalues as $\lambda_1\leq \lambda_2 \leq ... \leq \lambda_n $. Define the function as $f(A)=\sum_{i=1}^{k} \...
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1answer
82 views

Numerical Stability of a Generalized Spatial Discretization Scheme

After reading the matrix stability chapter (10) of Hirsch [1], I decided to dive in the reference list of the chapter. One of the papers [2], which is cited as reference shows an very interesting ...
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1answer
58 views

An Upper Bound of $\left<H,X\right>_F$ with Constrainted Rank

We have $X=[\mathbf{x}_1,\mathbf{x}_2...,\mathbf{x}_n]\in\mathbb{R}^{d\times n}$, $H=[\mathbf{h}_1,\mathbf{h}_2...,\mathbf{h}_n] \in\mathbb{R}^{d\times n}$, and $d<n$. $H$ has rank $r\leq d$ and $X$...
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223 views

Finding second excited state of Schrödinger equation with secant Runge Kutta method

In our assignment, we are required to find the energies of the ground state and the first two excited states of the Schrödinger equation in a harmonic potential: $$V = \frac{50 x^2}{(10^{-11})^2}\, .$...
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175 views

Can ARPACK exploit hermiticity when diagonalising a complex matrix?

I have noticed arpack comes with a driver dsdrv1 that exploits symmetry of a real-valued matrix. Is there a way to analogously exploit a Hermitian matrix in some way via z--- drivers? The manual ...

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