# 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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### 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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### How to use eigenvalue information to efficiently diagonalize matrices?

I apologize if this question in a more general form has been asked before. I have a tridiagonal Toeplitz matrix $K$, whose eigenvalues and eigenvectors are known analytically for any dimension $N$ ....
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### SLEPc eigensolvers take long time to converge for large sparse symmetric matrices

I am leveraging the SLEPc library for solving the first $k$ (where $k = 3$ or $4$) eigenvalues and their corresponding vectors for a matrix of size 200,000. The matrix is sparse and symmetric. I ...
203 views

### Generalized Eigenvalue Problem from linear stability analysis

I also posted this in the physics forum, but maybe here it fits better. I am trying to solve a generalized eigenvalue problem raised by linear stability analysis $$AV=\lambda BV.$$ $A$ and $B$ are ...
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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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### Estimating the spectral radius when the dominant eigenvalues are complex conjugates

I want to determine the spectral radius of a large non-symmetric matrix $A$ whose dominant eigenvalues are a pair of complex conjugates. My first instinct was to use a power iteration with a starting ...
161 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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### 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 ...
213 views

### Imposing boundary conditions for PDE quadratic eigenvalue problem

I have a quadratic eigenvalue problem of the form: $$(A_2 s^2 + A_1 s + A_0)\hat{v} = 0$$ where $s$ is the eigenvalue. The matrices $A_i$ contain derivatives up to order six, and I have six boundary ...
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### Singular matrix - but SVD works - what does the eigenvalues mean? Find I the “dependent” lines?

please can I ask a bit stupid question? I have a complex matrix A as a set of equations. I wanted to find the solution of Ax=b where b is vector of right-hand side. So I have called zgetrf on A (does ...
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### 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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### 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-...
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### Iteratively obtaining m eigenvectors using arpack: If I have a good initial guess, how do I use it?

I am trying out the arpack driver dsdrv1, which is used to iteratively obtain the first m eigenvectors from the eigenvalue problem. $$\hat{A}\mathbf{x} = \lambda\mathbf{x}$$ As it is an iterative ...
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### PageRank using Inverse Iteration Method by Cleve Moler

I was trying to understand how to use the inverse interation method to compute the page rank as an exercise. In this chapter (page 4) about page rank (by Cleve Moler), the author suggests to use the ...
484 views

### Laplacian discretization for parametric curves

I know how to compute the discrete Laplacian of a graph and of a mesh (the Laplace-Beltrami operator). Is there an analogous definition for the computation of the Laplacian of a parametric curve ? ...
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### 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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### How can QR iteration with complex matrices produce complex diagonal entries?

In Lapack (zhseqr) and matlab, the eigenvalues of a complex matrix are computed successfully. I notice that QR iteration or algorithm is involved with that process. QR iteration repeats to call QR ...
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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 decomposition of the sum: $AA^T$ + diag($u$)

Suppose $A\in\mathbb{R}^{n\times c}$,$u\in\mathbb{R}^n$,$n\gg c$. The time complexity of eigenvalue decomposing directly for matrix $AA^T+\text{diag}(u)$ is $O(n^3)$. And it is easy to avoid $O(n^3)$ ...