32 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-...
Federico Poloni's user avatar
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}_{...
GoHokies's user avatar
  • 2,186
15 votes

Why are systems with clustered eigenvalues easy to solve?

A good explanation of this phenomena with many examples is given in Iterative Methods for Linear and Nonlinear Equations by Tim Kelley. The crux of it comes down to the fact that each step of a ...
whpowell96's user avatar
  • 2,259
14 votes
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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 ...
Bill Greene's user avatar
  • 5,984
14 votes
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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 (...
Christian Clason's user avatar
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, ...
Kirill's user avatar
  • 11.4k
12 votes
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Real-world applications of eigendecomposition?

You can compute analytic functions of matrices using the eigendecomposition (or more generally by using the Jordan normal form in case the matrix is defective), you cannot do so with the singular ...
lightxbulb's user avatar
  • 1,994
11 votes
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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 ...
Federico Poloni's user avatar
11 votes
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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 ...
BrunoLevy's user avatar
  • 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}...
Kirill's user avatar
  • 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 ...
Federico Poloni's user avatar
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 ...
rchilton1980's user avatar
  • 4,862
10 votes
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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 ...
Federico Poloni's user avatar
9 votes
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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. <...
nicoguaro's user avatar
  • 8,490
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 ...
Kirill's user avatar
  • 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 ...
MPIchael's user avatar
  • 2,842
9 votes

Real-world applications of eigendecomposition?

In Quantum theory the observables corresponding to an operator are the eigenvalues of that operator. So, as an example, should you want the energy levels available to electrons in a molecule you need ...
Ian Bush's user avatar
  • 591
8 votes
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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 ...
Kirill's user avatar
  • 11.4k
8 votes
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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 ...
Bill Greene's user avatar
  • 5,984
8 votes
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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? $$...
GoHokies's user avatar
  • 2,186
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 ...
Federico Poloni's user avatar
8 votes
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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 ...
rchilton1980's user avatar
  • 4,862
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 ...
cbk's user avatar
  • 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 ...
Aage's user avatar
  • 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 &...
user3209427's user avatar
8 votes

Why are systems with clustered eigenvalues easy to solve?

At the $k$th iteration, typical Krylov methods for solving $Ax=b$ (such as CG, MINRES, and GMRES) implicitly construct a $k$th order polynomial $Q(x)$ such that: $Q(0) = 1$. $|Q(\lambda_i)|$ is as ...
Nick Alger's user avatar
  • 3,143
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} ...
Carl Christian's user avatar
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 ...
Kirill's user avatar
  • 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 ...
GoHokies's user avatar
  • 2,186
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 $\alpha$...
knl's user avatar
  • 2,076

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