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Questions tagged [eigensystem]

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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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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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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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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148 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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What strategies / decompositions would be useful to solve the following linear system repeatedly if I only care about time to solution?

Let $A\in \mathbb{R}^{n\times n}$ be symmetric positive semidefinite, and $B\in \mathbb{R}^{n\times n}$ be symmetric positive definite. Suppose $B$ is block diagonal so it is easy to invert. (We ...
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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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How to construct a eigensolver targeting a specific type of matrix

I need to diagonalize such kind of matrix during research: it's n-by-n, with it's upper-left (n-1)-by-(n-1) corner be diagonal while the nth row & column are dense. It's observed that the self ...
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Eigenvalues of a Laplacian operator on an irregular mesh

I have the following setting: An irregularly-shaped domain, expressed as a mesh of points A Laplacian operator, together with boundary conditions I am looking for the eigenvalues of that operator, i....
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Differential eigenproblem with eigenvalue in boundary condition

Statement of the problem I need to (numerically) solve an eigenproblem of the type $$-\omega^2\mathcal{D}_1\vec{x}=\mathcal{D}_2\vec{x}$$ on the interval $[-1,1]$, where $\mathcal{D}_1$ and $\...
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96 views

Time-stable spectral decomposition algorithm

Consider an $n \times n$ real, time-dependent matrix $A(t)$ such that $A(t) = A(t)^{T} > 0$ and $A(t)$ is continuous on $[a,b]$. Then it is posible to specify a matrix $S(t) \in SO(n)$ such that $S(...
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Left eigenvectors using ARPACK

I'm trying to find both the dominant $k$ left and right eigenvectors, that is, $$V_L\mathcal{A} = \Lambda V_L\\ \mathcal{A}V_R = V_R\Lambda\\ V_LV_R = I_{k\times k}$$ $V_L$ being the $k\times N$ ...
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Eigenvalue-style optimization with quadratic constraints

Suppose $A\in\mathbb{R}^{n\times n}$ is symmetric and positive definite and that we have several symmetric matrices $B_i\in\mathbb{R}^{n\times n}$ that are low-rank and indefinite. I need an ...
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272 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 \mathbf{...
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54 views

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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Scaling for a nonsymmetric eigenvalue problem

I have an eigenvalue problem emerging from the internal vibro-acoustic coupling. The eigenvalue problem is nonsymmetric but it was proven in literature that it results in real eigenvalues and ...
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110 views

Generalized eigenvalue with null space

Define $S\in\mathbb{R}^{n\times n}$ as $$S:=H+Q^\top V^{-1} Q.$$ $H,V$ are positive semidefinite. Here, $H$, $Q$, and $V$ are large, dense matrices but they are structured: I can write code for ...
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273 views

Numerical solution of Dirac equation (eigenvalue problem)

Suppose we have equation of the form: $$H \Psi = E \Psi $$ where $H$ is Dirac Hamiltonian (also my question can be answered by people who are not familiar with Dirac Hamiltonian but familiar with ...
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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: $A=P\Lambda_1 P^T$, $B=Q\...
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98 views

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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535 views

Perron-Frobenius theorem on general real symmetric matrices

From the Perron-Frobenius theorem, http://en.wikipedia.org/wiki/Perron%E2%80%93Frobenius_theorem#Perron_root_is_strictly_maximal_eigenvalue_for_positive_.28and_primitive.29_matrices it might be ...
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45 views

The sign of Schrodinger equation

I have a question for the format of Schrodinger equation $$\psi(x,t) = \int_0^\infty c_n e^{-iE_nt/\hbar} \psi_n(x)$$ Why do we have $i$ instead of $-i$?
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Methods to compute specific eigenvector components for a tridiagonal matrix

I have an application that is somewhat similar to the situation of computing Gaussian quadrature nodes and weights: simply put, I need to compute the eigenvalues and the last two (normalized) ...
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Solving 2D Schrodinger Equation with ARPACK: Can I ensure all eigenvectors have the same phase?

I use arpack to solve the 2D Schrodinger, and eigenvalue problem of the form $$Hx = \epsilon x$$ on a uniform grid. All eigenvectors are real in my case. Arpack doesn't normalise the eigenvectors, ...
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When numerically computing eigenstates during a coupled-mode-space NEGF calculation, do phases matter?

The coupled mode space NEGF method for computing transistor characteristics involves expanding the electronic wavefunction in a mode space basis $$\Psi(x,y,z) = \sum_n\phi_n(x)\xi_n(y,z;x)$$ where $\...
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44 views

Eigenvalue ODE in Spherical Coordinates--Numerical

I wish to solve an eigenvalue problem: $$\nabla^{2}f=Ef $$ If I assume spherical symmetry $f(r,\theta,\phi)=f(r)$, I can reduce the problem to 1D: $$(\frac{2}{r}\frac{d}{dr}+\frac{d^{2}}{dr^{2}})f=...
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142 views

How to make LAPACK eigenvectors orthogonal like Matlab?

I'm using LAPACK zgeev to calculate eigenvectors of a symmetric complex matrix of high dimensions ($n \approx 2000$). I need these eigenvectors to satisfy $$\sum_{...
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148 views

All eigenpairs of large sparse symmetric matrix

In advance I am sorry for my noobish question. I am a physics PHD student and basically I use python for my math/physics problems. But now I have a problem which requires more computing capacity and ...
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60 views

Coupled system of differential equations with eigenvalue

I have the following system of coupled differential equations (the Bogoliubov-de Gennes equations for a certain geometry): $$ \begin{cases} -\frac{1}{2}u''(r) + \big[\frac{1}{2r^2} + f_1(r)\big]u(r) +...
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156 views

Memory requirement to find eigenvalues and -vectors of large sparse matrix

How can I estimate how much memory will be needed to find eigenvalues and eigenvectors of a given large sparse matrix? I have a real symmetric matrix with roughly $5 \times 10^4$ rows and columns, ...
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27 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 \psi(x)=\...
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85 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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0answers
206 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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0answers
387 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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117 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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50 views

Size reduction of matrices in dispersion curve calculation

I have an energy dispersion curve obtained from the eigenvalues of $$E(k) = \text{eig}(T e^{ik} + T^H e^{-ik} + H_0),$$ where $H_0$ and $T$ are $N\times N$ square matrices, $T^H$ is the Hermitian ...