An eigenvector of an operator is a vector such that the action of the operator is the same as multiplication by a constant, called the eigenvalue. The eigensystem of an operator is the set of all such eigenvectors and their associated eigenvalues.

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Is there an efficient $O(n^2)$ way to get the eigen decomposition given a LDL factorization?

Let's say I have a LDL factorization of a matrix A. Is there an efficient $O(n^2)$ way to get the eigen decomposition of A given it's LDL factorization? Is there a more efficient way, in case L and ...
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Verification in Eigenvalue problems

Let us start with a problem of the form $$(\mathcal{L}u + k^2) u=0$$ with a set of given boundary conditions (Dirichlet, Neumann, Robin, Periodic, Bloch-Periodic). This corresponds with finding the ...
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106 views

How to correctly normalize modulus and phase of an eigenvector?

I am solving a linear stability problem using finite element discretization. Then, I have a generalised eigenvalue problem: $$ \lambda M x = J x.$$ I obtain complex eigenvalue and eigenvectors from ...
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14 views

Eigenvalue problem of the symmetric real operator which corresponds to the symmetric positive definite matrix

I have a real symmetric function $C(x,y)$ defined on $x,y\in[0,\infty)$, i.e. $C(x,y)=C(y,x)$. I want to solve the eigenvalues problem, i.e. find eigen values and eigen functions: $$\lambda ...
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30 views

Integro-differential eigenvalue problem, COMSOL

In my research I encounter an eigenvalue integro-differential equation of the form: $$f_n(x,y)=\lambda_n\iint_D\frac{\nabla'\cdot\big\lbrace ...
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45 views

large eigenvalues with LAPACK

I have question about LAPACK. I calculate eigenvalues of a $16\times16$ Hermitian complex matrix with small entries by ZHEEV ...
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75 views

Armadillo eig_sym() for extracting eigenvalues. Is it parallel at all? [closed]

After wasting 3 days with scalapack, I gave up and moved to Armadillo, considering it uses lapack underneath its beatiful and easy interface. I would like to calculate the eigen values and eigen ...
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56 views

Iteratively finding both left and right eigenvectors for non-symmetric complex matrix

I have a complex, non-Hermitian matrix $\mathbf{A}$, for which I need to find a few eigenvalues and eigenvectors in the generalised eigenvalue problem: $$\mathbf{A}\cdot \mathbf{x} = \lambda ...
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85 views

How to choose (and many) threshold in Eigenface?

I edited my question trying to make it as short and precise. I am developing a prototype of a facial recognition system for my Graduation Project. I use Eigenface and my main source is the document ...
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56 views

most efficient way to calculate eigen states of a 2D or 3D potential (Matlab)

I know of several ways to calculated the eigen states of 1D potentials (i.e. DVR, Crank–Nicolson, etc). However I wonder what is the most efficient way to do the same for a N-Dimensional potential? ...
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68 views

Compute left and right eigenvectors in c++ using Eigen library

The zgeev routine of the Lapack library computes the eigenvalues and left and/or right eigenvectors of a given matrix. I want to use c++ and the eigen library to do the same. A ...
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136 views

How to impose boundary conditions on eigenfunction problems?

I am trying to solve for the eigenfunctions of a (1D) differential operator using finite differences: $$A \, f(x) = \lambda f(x)$$ Here is an example in Python where $A = \partial_x^4$: ...
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Parallel Monte Carlo simulation using PETSc

I am trying to do Monte Carlo simulation for a large problem which requires eigensolution of a matrix for each sample. The matrix itself is quite large so much so that I want the eigensolution itself ...
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55 views

Computing eigenpairs of singular matrix with ZGEEV?

I've never run into a singular matrix before, so bear with me. I have a complex non-symmetric matrix (about 1000 x 1000) that I know has a couple zero eigenvalues. It isn't guaranteed to be ...
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108 views

diagonalization of matrix - omitting small matrix elements

I was wondering whether there is some theorem that allows me to put an upper bound on the error introduced by omitting small matrix elements from a matrix before diagonalization. Let's assume we ...
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74 views

Numerical eigenbasis for a unitary matrix

Do you know what numerical software computes an eigenvector basis for a unitary matrix? Say I have a unitary matrix $U$. If its eigenvalues are simple (no multiplicities), then for instance Matlab ...
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211 views

calculating eigenvector components of a given vector

I have some vector $V$ which can be decomposed into the eigenspace of the hermitian sparse operator $M$: $V = \sum_i v_i \hat{m}_i$ Is there a way to find the $\hat{m}_i$ (the eigenvector itself) ...
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95 views

roots of polynomials of high degree: LinAlgError: Eigenvalues did not converge

I wrote a simple script to generate random polynoimals $\displaystyle f(z)= \sum_{k=0}^N a_k \frac{z^k}{\sqrt{k!}} $ of high degree and find their roots. For more discussion on random polyomials see ...
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45 views

LAPACK DGGEVX: BALANC option

I'm using DGGEVX routine from LAPACKE with BALANC option as shown below, but to my surprise changing BALANC option from 'N' to ...
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154 views

I am looking for a complex sparse matrix EigenVector solver for GPGPU; preferably CUDA

So far the closest I've found is ViennaCL, which has a Lanczos implementation for Eigenvalues. It is not clear that EigenVectors are produced by this library. Does anyone here know whether ViennaCL ...
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87 views

Eigen vector property: MATLAB 'chol' (LAPACK DSYGV) & MATLAB 'qz' (LAPACK DGGEVX)

Two Eigen algorithms return Eigen vectors with different properties: 1st algorithm, LAPACK DSYGV (the same as MATLAB eig with ...
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2answers
464 views

Eigenvectors: MATLAB vs LAPACK DGGEV or DGGEVX

If we call LAPACK DGGEV or DGGEVX routines for two badly-conditioned matrices in a C++ code, will we get the same eigen-values ...
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Fortran 2003 ARPACK wrapper

I wrote a Fortran 2003 wrapper for the ARPACK routine znaupd, basically translating the the example driver routine zndrv1 into modern Fortran 2003 language with automatic arrays. I initialize every ...
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531 views

Most efficient library to diagonalize exactly large hermitian or unitary matrices

I am working on a physics problem which requires obtaining the exact eigenvalues and eigenvectors of Hermitian and Unitary matrices numerically. Naturally I would like to ask the experts what are the ...
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326 views

Comparing Eigenvectors, Mathematica vs. Matlab

I am trying to create the same outputs in Mathematica and Matlab, however I am running into trouble aligning the eigenvectors with the eigenvalues, I think the Matlab is doing something slightly more ...
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210 views

Sparse smallest eigenvalue problem on a linear subspace?

I am interested in methods for solving the optimization problem $$ \begin{array}{rl} \arg\min_x & x^T A x \\ \mathrm{s.t.} & x^T x = 1 \\ & Bx = 0 \end{array} $$ where $A$ is symmetric ...
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87 views

Finding Interior eigenvalues using Davidson algorithm

Is it possible to find interior eigenvalues closer to some lambda using Davidson method. I was searching online but found that most people use Jacobi-Davidson method for that. Thanks
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Computing eigendecomposition of a Hermitian matrix that is almost unitary

I have a dense Hermitian matrix that is approximately unitary, so it has eigenvalues that are $\sim \pm1$. I would like to compute all the eigenvectors corresponding to the $+1$ eigenvalue (not ...
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208 views

How to find the smallest positive eigenvalue of a large general system if they are all in +/- pairs of real eigenvalues

I used MKL Lapack SBGVX because it can solve for "selected" eigenvalues/modes (both positive and negative), thinking it would be efficient, but it is extremely slow when compared with Bathe's subspace ...
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1answer
53 views

last column of SPD matrix given it's spectral decomposition

I'm working on this application where I get the spectral decomposition (O,D) of a matrix A for free (A is of full rank) but not A itself, and I need to only recover the last column of A. I was ...
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Dominant contributions of a quadratic form

Let $\Sigma$ be a covariance matrix (e.g. symmetric positive definite). For arbitrary vectors $\epsilon$, I need to compute $\chi^2 \equiv \epsilon^\top\Sigma^{-1}\epsilon$, which I do using a ...
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84 views

Locally conservative method for differential generalized eigenvalue problem

I have to approximate the smallest eigenvalue of the following generalized eigenvalue problem $$ - \nabla \cdot D(x) \nabla p(x) + \alpha(x) p(x) = \lambda \beta(x) p(x) $$ over a domain like ...
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Is there guaranteed global solver for such an eigenvalue problem?

The original nonlinear optimization problem I have is as follows: For constant symmetric matrices $A=A^T, B_i=B_i^T(\forall i\in\mathbb{N}) \in \mathbb{R}^{n\times n}, \text{rank}(A)=n,$ ...
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Algorithm for directly finding the leading eigenvector of an irreducible matrix

According to the Perron-Frobenius theorem, a real matrix with only positive entries (or one with non-negative entries with a property called irreducibility) will have a unique eigenvector that ...
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Are the eigenvalues of the product matrix of two real symmetric square matrices also real values?

Suppose $A,B \in \mathbb{R}^{n\times n}; A=A^T, B=B^T$, let $C = AB, D =BA$, If we have all the real eigenvalues of $A$ and $B$, e.g. the eigenvalue decomposition of them: $A=P\Lambda_1 P^T$, ...
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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 ...
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1answer
167 views

Algorithm for Eigenvalue Problem of a Real Symmetric nxn Matrix

I have a nxn covariance matrix (so, real, symmetric, dense, nxn... I mean positive semi-definite). 'n' may be very very very big! I'd like to solve partial (3 largest) eigenvalue (+eigenvectors) ...
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288 views

Algorithm for Complete Eigenvalue Problem of a Real Symmetric nxn Matrix

I have a nxn covariance matrix (so, real, symmetric, dense, nxn). 'n' may be very very very big! I'd like to solve complete eigenvalue (+eigenvectors) problem for this matrix. Could somebody tell me ...
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307 views

Implementation of Jacobi-Davidson method for cubic eigenvalue problem

I have a large cubic eigenvalue problem: $$\left(\mathbf{A}_0 + \lambda\mathbf{A}_1 + \lambda^2\mathbf{A}_2 + \lambda^3\mathbf{A}_3\right)\mathbf{x} = 0.$$ I could solve this by converting to a ...
2
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1answer
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How to implement the spectral decomposition of a symmetric dense matrix via Eigen C++

Spectral decomposition of symmetric matrix $A_{n\times n}$, specifically, $n=3$ find the orthogonal matrix $Q$ and diagonal matrix $\Lambda$ such that: $A=Q\Lambda Q^T$ How to implement such ...
2
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2answers
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About Subspace Iteration for Eigenvalues

I heard that subspace iteration plus Ritz acceleration could improve the performance a lot for solving clustered eigenvalues, for the eigenvalues and eigenvectors could converge linearly with ratio ...
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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 ...
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Solver for eigensystem of vectors?

I'm trying to solve a multipole system. It involves a matrix of 3x3 tensors $A_{ij}$ and a vector of 3-tuples $\mathbf v_i$. $$\left(\begin{matrix} A_{11} & A_{12} & \cdots & A_{1n}\\ ...
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Size reduction of matrices in dispersion curve calculation

I have an energy dispersion curve resulted from eigen values of E(k) = eig(T exp(ik) + T' exp(-ik) + H0). Where H0 and T are NxN square matrices and T' is transpose of T and k is wave-vector. So there ...
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What is the added cost of generalizing an eigensystem?

Problem Let's say I can write a model as the Hermitian eigensystem: $$ A x = \lambda x $$ where $A \in \mathbb{C}^{n\times n}$ is Hermitian, or as the generalized Hermitian eigensystem: $$ \tilde A ...
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Does Lanczos have trouble with large matrix elements?

I have a large, yet very sparse, matrix that I'd like to diagonalize. Both my own Lanczos implementation and the ARPACK that's built in with scipy fail to converge properly, though. I know that my ...
2
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how to use lanczos code from netlib for large sparse symmetric matrix?

I want to use lanczos method to calculate the few lowest eigenvalue and eigen-vector of a large sparse symmetric matrix(~50k x ~50k). In http://www.netlib.org/lanczos/index.html I found the codes ...
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229 views

Preconditioner for finding the smallest eigenpairs of a large, but structured, matrix

I'm trying to find the eigenvector corresponding to the second smallest eigenvalue of a large $(4,000,000 \times 4,000,000)$ matrix $L$. $L$ is a graph Laplacian, with the following structure: $L = D ...
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Preconditioner for large size hermitian eigenvalue problems

Basically I try to compute several smallest eigenvalues of some sparse 50k*50k eigenvalue problems using matlab. $$Ax = \lambda Bx$$ With matlab eigs, it's not as fast as I expected. So I tried some ...
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253 views

Continuity of eigenvectors of parametric matrix

I have $n$-dimensional matrices $\mathrm{\hat{H}}(\vec{k})$ depending on vector parameter $\vec{k}$. Now, eigenvalue routines return eigenvalues in no particular order (they are usually sorted), but ...