# Tag Info

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-...
• 11.5k

### 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 ...
• 2,636
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 ...
• 6,144
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 (...
• 12.3k
Accepted

### 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 ...
• 2,197
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 ...
• 2,315
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

• 2,216

### 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 ...
• 2,985

### 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 ...
• 616

### Optimized Lanczos method for finding eigenvalues of $A \otimes B$

There isn't much complicated behind this idea; it's just that since Lanczos is a black-box method you can use any method of your choice to compute the products $v\mapsto (A\otimes B)v$ needed in the ...
• 11.5k
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
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 ...
• 6,144
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 ...
• 11.5k
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 ...
• 4,936

### 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
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
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

### 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 “...
• 2,761

### 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 ...
• 3,143

### 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 $$\lambda_1 \leq \lambda_2 \leq \lambda_3 \leq \lambda_4$$ ...
• 1,391
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
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 ...
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 $\alpha$...