31
votes
Accepted
Why do we usually not want the eigenvalues of non-symmetric matrices?
Stability under perturbations
Let $E$ be a perturbation such that $\|E\| \leq \varepsilon$.
If $A$ is symmetric, then the eigenvalues of $A+E$ are at a distance $\varepsilon$ from those of $A$. (Bauer-...
- 10.6k
15
votes
Accepted
Smallest eigenvalue without inverse
Compute the largest-magnitude eigenvalue $\lambda_{max}$ of $A$ (with, say, eigs('lm')).
Then compute the largest magnitude (negative) eigenvalue $\hat{\lambda}_{...
- 2,166
14
votes
Accepted
Quality of eigenvalue approximation in Lanczos method
The convergence behavior you are seeing is actually expected. One of
things that makes the Lanczos method so interesting is that it does
a good job of simultaneously converging eigenvalues at both ...
- 5,934
13
votes
Accepted
How does the QR algorithm applied to a real matrix returns complex eigenvalues?
In a nutshell, the QR algorithm applied to a matrix $A$ is an iterative procedure that converges to the real Schur decomposition: a unitary matrix $Q$ and a matrix $R$ in block upper triangular form (...
- 12.1k
12
votes
Accepted
Eigenvalues of Small Matrices
The first thing to note is that the correspondence between finding
roots of a polynomial (any polynomial) and finding the eigenvalues
of an arbitrary matrix is really direct, and it's a rich subject, ...
- 11.4k
11
votes
Accepted
Compute all eigenvalues of a very big and very sparse adjacency matrix
You can use the shift-invert spectral transform [1] and compute the spectrum band by band.
The technique is also explained in my article [2]. Besides the implementation in [1], an implementation is ...
- 2,305
11
votes
Complex Eigenvalues using eig (Matlab)
You have an ill-conditioned eigenvalue problem. Consider a perturbation $\delta A$ in your matrix $A$—with double-precision floats this is $O(10^{-16})$. It turns the eigendecomposition from
$$ X^{-1}...
- 11.4k
10
votes
Accepted
How does gmres method iteration behave for this non-diagonalizable matrix?
Unfortunately, convergence of GMRES does not have a clear dependence on the distribution of eigenvalues.
It was proved by Greenbaum, Ptak and Strakos in 1996 that you can construct examples with an ...
- 10.6k
10
votes
Cheap recalculation of eigenvalues and eigenvectors for a low-rank update of the matrix
Unfortunately, I don't think there is a good algorithm to do this efficiently.
Given the eigendecomposition $\mathbf A = \mathbf X \mathbf D \mathbf X^T$, one is tempted to project $\mathbf v$ onto ...
- 4,794
10
votes
Accepted
Why are all eigen solvers iterative?
There is simply no closed-form expression in terms of the four operations and radicals for the eigenvalues of a matrix greater than $4\times 4$.
This follows from the facts that (1) there are ...
- 10.6k
9
votes
Accepted
Correct eigenfunctions of Laplace operator by Finite Differences
You should specify the eigenvalues you want with which="SM", for example.
Check the following snippet. I also changed the solver, since your system is symmetric.
<...
- 8,312
9
votes
Nonlinear eigenvalue problem - MATLAB code
Given a nonlinear eigenvalue problem of the form $A(\lambda)x = 0$, reducing it to a real equation $\det(A(\lambda))=0$ is known to be a poor method for just the reason you've discovered yourself. The ...
- 11.4k
9
votes
Is LAPACK behind the cutting edge of dense linear algebra?
When one says an algorithm is of order $O(n)$, that may mean that the complexity is given by: $c + b*n$. With every new element you add you increase in runtime (effectively). What mathematically ...
- 2,621
8
votes
Accepted
Stabilizing a 3x3 real symmetric matrix eigenvalue calculation
This is trying to compute the eigenvalues by computing the roots of the characteristic polynomial. In this case, the characteristic polynomial is $p(t) = t^3-2t^2x$, $x=1.25\times 10^6$, and zero is a ...
- 11.4k
8
votes
Accepted
Eigen - Max and minimum eigenvalues of a sparse matrix
There are two relatively convenient options for calculating
selected (e.g. a few largest or smallest) eigenvalues using Eigen.
The first is Spectra, a header-only C++ library based on Eigen
that uses ...
- 5,934
8
votes
Accepted
Convexity of Sum of $k$-smallest Eigenvalue
Given $A \in {\bf S}^n$ (a positive definite matrix) with eigenvalues $\lambda_1 \leq \lambda_2 \leq \ldots \leq \lambda_n $, then:
$\displaystyle f_k(A)=\sum_{i=1}^{k} \lambda_i$ is concave. Why?
$$...
- 2,166
8
votes
Accepted
How to compute all the eigenvalues of a large sparse matrix using matlab?
"Get more RAM" may be one of your best options. :) Prices are reasonably low right now, and it's one of the best upgrades you can gift your computer anyway. 10k x 10k is borderline but still doable ...
- 10.6k
8
votes
Accepted
Implementation of a $O(n \log(n))$ method to compute eigenvalues of real symmetric tridiagonal matrices
I think the method has too much implementation complexity and too narrow applicability to be worth it.
Though the paper is correct to point out the importance of solving the tridiagonal-symmetric ...
- 4,794
8
votes
Is LAPACK behind the cutting edge of dense linear algebra?
LAPACK has been on the cutting edge for just about three decades, and probably still is for its niche. However, given given recent developments in libraries for the simpler BLAS-type matrix operations ...
- 181
8
votes
Accepted
Can this simple quadratic optimization problem be turned into a simple eigenvalue problem?
At least for the second question the answer is yes. See for example Mattheij, Robert MM, and Gustaf Söderlind. "On inhomogeneous eigenvalue problems. I." Linear Algebra and its Applications ...
- 188
8
votes
Accepted
Python scipy eigh(Arpack) giving wrong eigenvalues for generalized eigenvalue problem
The matrix B (M in the documentation) needs to positive definite according to the documentation: "If sigma is None, M is positive definite", this is in addition to the first requirement &...
- 407
7
votes
Is there an eigenvalue estimation method more accurate than Gershgorin's, which uses no multiplication?
Conclusions are indeed possible, but Gershgorin's circle theorem must be supplemented with other results. Let
\begin{equation}
\lambda_1 \leq \lambda_2 \leq \lambda_3 \leq \lambda_4
\end{equation}
...
- 1,380
7
votes
Accepted
Eigenvalues of $ab^T$
This is sometimes known as Buzano's inequality (http://www.jstor.org/stable/2159168). In general, if $\|x\|=1$, $P=xx^{\top}$ is a projection operator, so a simple application of Cauchy-Schwarz leads ...
- 11.4k
7
votes
Eigenvectors of a small norm adjustment
There exist special techniques for updating the eigen-decomposition of time-dependent covariance matrices. Given a "prior" eigenvalue decomposition (say at some initial time $t^0$), these recursive ...
- 2,166
7
votes
Matlab, Mathematica & LAPACK returning 3 different eigenvectors
Seems that you have a duplicate eigenvalue. Thus, you have two eigenpairs $(\lambda_1, x_1)$ and $(\lambda_2, x_2)$ where $\lambda_1 = \lambda_2$. Denote $\lambda = \lambda_1 = \lambda_2$. Let $\alpha$...
- 2,041
7
votes
Accepted
Eigenvector with maximum overlap
The following paper suggests that the Jacobi-Davidson method can be used to target eigenvectors based on "any property that can be computed from the eigenvector", which would seem to include overlap ...
- 303
7
votes
Accepted
Numerically find the nearest positive semi definite matrix to a symmetric matrix
After you compute $Q$ and $D$, form $D'=\max(D,0)$, and compute $A'=QD'Q^\top$, the algorithms involved in multiplying those matrices do not promise that $A'$ will be exactly $QD'Q^\top$. Most ...
- 11.4k
7
votes
Why do we usually not want the eigenvalues of non-symmetric matrices?
Amazing question which has a long answer, but I will try to be concise. In the context of Krylov subspace methods for general matrices, the eigenvalues of a non-symmetric matrix mean very little. In “...
- 2,321
6
votes
Accepted
Sparse generalized eigensolver using OpenCL
One possibility is to use a combination of ARPACK and ViennaCL:
ARPACK is an eigensolver. It works with a callback interface (you supply a function that computes $Ax$ for a given $x$ and it computes ...
- 2,305
6
votes
Accepted
What is the state of the art algorithm for diagonalizing real symmetric matrices?
The QR/Francis algorithm is the go-to choice for dense eigenproblems, but there are a few competitors around:
The Jacobi algorithm (like QR, another algorithm with an unfortunate name, which can be ...
- 10.6k
Only top scored, non community-wiki answers of a minimum length are eligible