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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4
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0answers
76 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
106 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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23 views

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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54 views

Bound for Expectation of Singular Value

In my case, $X_{\boldsymbol{\delta}}\in\mathbb{R}^{d\times M}$ is a function of Rademacher variables $\boldsymbol{\delta}\in\{1,-1\}^M$ with $\delta_i$ independent uniform random variables taking ...
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2answers
67 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
59 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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46 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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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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100 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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102 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 ...
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56 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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1answer
83 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
82 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
102 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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67 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 ...
2
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1answer
124 views

A Bound for the inverse of the sum of identity and triangular matrix

I wonder if there are any theorems which can help me to calculate an upper bound for the spectral norm of: $$\left\| \left[ I + \sum_{i=1}^{\overline{n}\in\mathbb{N}} \big( C_i - I\big)\right]^{-1}\...
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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
119 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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3answers
338 views

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 ...
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3answers
162 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
69 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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135 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\...
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2answers
248 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
95 views

Benchmark problems for eigenvalue reordering algorithms sought

Every real matrix $A$ can be reduce to real Schur form $T = U^T A U$ using an orthogonal similiary transform $U$. Here the matrix $T$ is quasi-triangular form with 1 by 1 or 2 by 2 blocks on the main ...
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1answer
101 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 ...
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1answer
186 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 ...
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1answer
73 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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56 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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3answers
262 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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3answers
792 views

Efficient eigen-decomposition of covariance matrix

I am looking for an C/C++/Python algorithm implementation that calculates eigenvalues and eigenvectors of a symmetric, positive semidefinite covariance matrix. A general-purpose eigen-decomposition ...
2
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1answer
91 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
39 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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0answers
61 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 ...
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1answer
126 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 ...
4
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1answer
245 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
55 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 ...
2
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1answer
87 views

Taylor expansion of error - Finite elements approximation

In some of my computations I calculate a scalar value $\lambda_h$ (in my case an eigenvalue) depending on a finite element discretization of the domain. Usually we can manage to find estimates of the ...
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1answer
55 views

LAPACK non-convergent eigenvalue algorithm for complex but not double matrix

I've encountered an odd issue with solving for the eigenvalues of the following matrix, in Matlab format: ...
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1answer
87 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 ...
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0answers
42 views

What is the complexity of calculating K-th largest real part eigenvalue of non-normal sparse matrix

I just need to calculate the largest real part of eigenvalues of a Jacobian which is highly non-normal and singular. Most of the eigenvalues are negative, and some of them are positive but near to ...
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0answers
99 views

What is Chebfun `eigs` doing

What is this doing? Looks like the original eigenvalue problem is converted into generalized eigenvalue problems with different dimensions of collocation points. Can someone explain more about this? ...
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94 views

Low memory algorithm for matrix diagonalisation

I'm trying to find the largest eigenvalues of very large $N \times N$ matrices ($N = 10^{10}$ and larger). The matrices are not sparse but the multiplication operation is fast. For now, I'm using ...
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245 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, ...
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1answer
234 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 ...
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0answers
46 views

Appropriately handling boundary conditions in a PDE eigenvalue problem

Suppose I have a nonlinear second order Cauchy PDE $\dfrac{\partial p(x,t)}{\partial t}=N(p(x,t))$, where $N:L^2(\mathbb{R})\rightarrow L^2(\mathbb{R})$, and a known fixed point $u(x)$. Mathematically,...
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0answers
149 views

Multi-matrix orthogonal basis problem

Suppose we are given a set of symmetric, positive definite matrices $A_1,A_2,\ldots,A_k\in\mathbb{R}^{n\times n}$. Is there any numerical method or reduction to a known problem (e.g. eigenvalue ...
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2answers
219 views

Eigenvalues of Small Matrices

I'm writing a small numerical library for 2x2, 3x3, and 4x4 matrices (real, unsymmetric). A lot of numerical analysis texts highly recommend against computing the roots of the characteristic ...
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2answers
178 views

Computing Eigenvectors in MATLAB

I am assigned to compute eigenvalues and eigenvectors in MATLAB of a 2x2 matrix: $$ A = \left( \begin{matrix} 3 &0\\ 4 &5\\ \end{matrix} \right) $$ I know that the textbook's solution states ...
2
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2answers
481 views

Complex Eigenvalues using eig (Matlab)

I wanted to find and plot the eigenvalues of large matrices (around1000x1000). But discovered when using the eig function in matlab, it gives complex eigenvalues when it shouldn't. For example, in the ...