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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1answer
205 views

Eigenvectors associated to two quasi-degenerate eigenvalues

I need to find the smallest eigenvalue and the corresponding eigenvector of a sparse matrix $M$ whose dimension is $\approx 10^4$. Within Matlab enviroment, I use the command ...
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1answer
235 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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682 views

Nonlinear eigenvalue problem - MATLAB code

I'm trying to solve a nonlinear eigenvalue problem in MATLAB, still without success. It's a problem about graphene plasmonics. The nonlinear eigenvalue problem is given below: \begin{equation} \frac{...
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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
131 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 ...
3
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1answer
103 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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1answer
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How to use Lanczos method to compute eigenvalues and eigenvectors

I have a sparse and symmetric matrix A(n x n). The method Lanczos tranforms matrix A into tridiagonal and symmetric matrix T and the Lanczos vectors in matrix V. From there how do I compute k ...
4
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2answers
151 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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3answers
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How does the QR algorithm applied to a real matrix returns complex eigenvalues?

I'm a noob into eigenvalues algorithms, but something call my attention. QR algorithm works with real/complex matrices producing real/complex eigenvalues. However, it can not produce complex ...
2
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1answer
209 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 ...
4
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1answer
71 views

Solver for generalized eigenvalue problem with multipoint constraints

We have the following generalized eigenvalue (set of) problem(s) $$[K_R(\kappa)]\{u_R\} = \omega^2[M_R(\kappa)]\{u_R\}\quad \forall \kappa \in [\kappa_0, \kappa_1]$$ with \begin{align} &K_R(\...
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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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1answer
244 views

Use SLEPc from Matlab

Is there a direct way to use SLEPc from Matlab? I remember that in some old manuals there was some Matlab interface. However, in the last one, I cannot find any reference to this. For me, it would be ...
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2answers
468 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
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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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234 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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2answers
192 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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1answer
160 views

Finding the $i$-th largest eigenvalue of a matrix

Given a large matrix $A$ with eigenvalues $\sigma_1\ge \sigma_2 \ge \dotsc $, I want to determine only a subset of these values, say $\sigma_5,\sigma_8$ and $\sigma_{19}$. Is there an algorithm that ...
6
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1answer
355 views

Dirichlet boundary conditions in generalized eigenvalue problem

Let us consider a problem of the form $$(\mathcal{L} + k^2) u(\mathbf{x})=0\, ,\quad \forall \mathbf{x} \in \Omega$$ with Dirichlet boundary conditions $$u(\mathbf{x}) = 0, \quad \forall \mathbf{x} ...
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1answer
373 views

Eigenvector with maximum overlap

Given a matrix $M$ and a vector $v$, is there an efficient method to find the normalized eigenvector of $M$ that is closest to $v$, in that it has maximal overlap. More explicitly, a vector $v$ can be ...
4
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1answer
603 views

Numerically find the nearest positive semi definite matrix to a symmetric matrix

I have a symmetric matrix $M$ which I want to numerically project onto the positive semi definite cone. To do so, I decompose it into $M = QDQ^T$ and transform all negative eigenvalues to zero. (...
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1answer
160 views

Numerical solution of two coupled nonlinear eigenvalue problems

I would like to numerically solve the following system of coupled nonlinear differential equations: $$ -\frac{\hbar^2}{2m_a} \frac{\partial^2}{\partial x^2}\psi_a + V_{ext}\psi_a + \left( g_a |...
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1answer
649 views

Estimate extreme eigenvalues with CG

CG may be used to estimate the extremal eigenvalues of a SPD matrix (by computing eigenvalues of tridiagonal matrix associated with the Lanczos algorithm). After a few iterations the largest ...
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0answers
79 views

Improving convergence of Jacobi iteration to Schur form

I'm using SIMD processor arrays to compute the eigen-decomposition for large numbers of small (up to $32\times 32$) matrices. For assorted technical reasons, Jacobi iteration maps well to the SIMD ...
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0answers
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Why the MIRACLE of Lanczos/CG-like?

Lanczos/Arnoldi/Rietz/CG-like algorithm share the same core strategy... In each, a little miracle appears, most of the Gram-Schmidt inner products are zeroes! In others words, new direction need only ...
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1answer
878 views

Sorting eigenvalues by the dominant contribution

[Edited to simplify the question] I am trying to associate the eigenvalues $E$ of a matrix $H$ to the original rows of the matrix. Moreover, it would be trivial to sort the eigenvalues in ascending ...
3
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3answers
336 views

Derive the formula for eigenvalues

If $A$ has eigenvalue $\lambda_A$ $$B = I - c\frac{I-rA}{I-\bar{r}A}$$ How to derive the eigenvalue $\lambda_B$? $$\lambda_B=1-c\frac{1-r\lambda_A}{1-\bar{r}\lambda_A}$$ where $c, r, \bar{r}$ are ...
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Solving a coupled eigen value problem

I have the following problem: $$\begin{bmatrix}A &B \\C& D\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix} = \begin{bmatrix}\lambda I_m & 0 \\ 0& \mu I_n\end{bmatrix}\begin{bmatrix}x \\y\...
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1answer
126 views

Computing real normal modes from complex eigenvectors

I'm trying to get the normal modes of a system of springs and dasphots using the basic dynamic equations for a linear, damped elastic structure: $ M \ddot{u}(t) + C \dot{u}(t) + K u(t) = f(t) $ to ...
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0answers
260 views

Computing Small Eigenvalues with Sparse Symmetric Indefinite Mass Matrix

I want the eigenvalues of the following generalized eigenvalue problem: $$ Av = \lambda M v $$ where $A\in\mathbb{R}^{n\times n}$ is sparse, symmetric, and positive semi-definite $M\in\mathbb{R}^{n\...
6
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2answers
266 views

inertia count sparse matrix with dense low-rank perturbation

I would like to determine the number of negative eigenvalues (inertia count) of the $(N \times N)$ symmetric real matrix $K - \sigma M$, with $K$ a positive-definite sparse matrix and $M$ a positive-...
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1answer
474 views

Eigenvalue problem with periodic boundary conditions: Are my eigenvalues correct?

I am using a (central) finite difference scheme to solve the eigenvalue problem $$-\frac{d^2}{dx^2}u = \lambda u$$ with periodic boundary conditions on a unit interval. I use arpack's zndrv1 and ...
3
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1answer
291 views

Compute nearly degenerated eigen values and vectors

In a quantum physics calculation, I have to deal with a matrix that has a lot of eigenvalues really close to each other ($10^{-6}$ relative difference for example) and I can't manage to obtain ...
4
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1answer
103 views

Compute specific eigenvalues in the complex plane with Feast?

In physical problems, it's quite common that we need to solve for specific eigenvalues in the complex plane, e.g. with a positive real part and negative imaginary part. In this case, we are looking ...
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1answer
135 views

Numerical solution for eigenvectors and eigenvalues of a Sturm-Liouville problem

I have to deal with the following problem in my research: $$\left[\frac{1}{D}F_{x}\right]_{x} + \frac{f D_{x}}{c D^{2}}F = 0$$ with boundary conditions $$F(0) = 0$$ $$F_{x}(L) = 0$$ where $f$ is ...
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1answer
255 views

FEM or FD eigenvalue equation to get wave number instead of cutoff frequency

To get cutoff frequencies and eigenmode field distributions for a waveguide, one uses following equation: $$1/\epsilon ∇ \times 1/\mu ∇ \times E = \omega^2 E$$ With $ \omega^2 $ as eigenvalues. This ...
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0answers
58 views

Geometric interpretation of lemma

I am currently studying eigenvalue problems. I already worked through the Minimax-principles, seen why $\lambda_{h, m} \geq \lambda_m$, when comparing the eigenvalues of a discretization and the ...
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346 views

Eigenvalue with largest imaginary part

Iterative eigensolvers such as ARPACK, give the option to find a subset of the eigenvalues which have the largest imaginary part. My question is how do these algorithms work. As I understand it, ...
3
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2answers
1k views

Stabilizing a 3x3 real symmetric matrix eigenvalue calculation

I have many 3x3 real symmetric matrices for which I need to determine the eigenvalues. Wikipedia gives a nice non-iterative algorithm for this case, which I have translated into C++: ...
2
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1answer
222 views

how to calculate the determinant of a projection of matrix to a subspace

I have a matrix $J$, and I know there are 3 existing zero eigenvalues and their eigenvectors. I want to detect if there is one extra eigenvalue to go cross zero (if it is zero, its eigenvalue will ...
3
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3answers
205 views

Eigenvectors of Black-box matrix

$\DeclareMathOperator{\diag}{diag}$ Consider the generalized eigenproblem $A\mathbf{x}=\lambda B\mathbf{x}$. When solving PDEs numerically (specifically, I am interested on finding the Dirichlet ...
6
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2answers
2k views

Numerical Solution to Schrödinger Equation--Multiple Wells

I am trying to solve for the allowed wavefunctions and energies for a 1D quartic potential well. To do this I am using the patching method (https://engineering.dartmouth.edu/microeng/otherweb/...
2
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1answer
117 views

FEM-Laplace with Dirichlet in only a few points: Nonsingular operator?

Let's consider the FEM discretization of the Laplace operator without boundary conditions, i.e., $$ a(u,v) = \int_\Omega \nabla u \cdot \nabla v - \int_\Gamma (n\cdot \nabla u) v. $$ For one-...
2
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1answer
50 views

Partial diagonalisation of large symmetric positive-definite band-diagonal matrices

I want to partially diagonalise real sparse symmetric positive-definite matrices, that are of dimension $n = 10^5$ and I need on the order of $k = 500$ of the smallest eigenvalues and eigenvectors. ...
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1answer
144 views

Finding the lowest $n$ eigenvalues of a band-diagonal Matrix

I have a real sparse matrix of the form $$ \left( \begin{array}{ccc} h_{11} & h_{12} & 0 & h_{14} & & & \\ h_{21} & h_{22} & h_{23} & 0 & h_{25} & & ...
2
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1answer
703 views

eigs routine in octave [closed]

I am using octave and observed a problem with the eigs-routine for non symmetric matrices. Using GNU octave version 3.8.1 the code below gives significant difference of eigenvalues although same ...
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0answers
87 views

Finite differenced eigenvalue prob. of inhomogeneous boundary conditions?

I am basically asking about eigenvalue problems of differential equations using some finite difference method (FDM). Usually the system is subject to some boundary conditions (BC), e.g., Dirichlet or ...
4
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1answer
463 views

Nystrom approximation of SVD for asymmetric matrices

Suppose I have a symmetric matrix $K$. Subdivide $K$ into pieces as $$K=\begin{pmatrix} K_{11} & K_{12} \\ K_{21} & K_{22}\end{pmatrix},$$ where $K_{21}=K_{12}^\top$. Then, the Nystrom ...
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1answer
59 views

LAPACK zlapmt_ freezes code [closed]

I'm using LAPACK's zggev_ routine to solve some generalized eigenvalue problem. While it produces the correct results, I want the eigenvalues and according eigenvectors sorted by absolute value. For ...