# 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).

239 questions
Filter by
Sorted by
Tagged with
438 views

### Applying Dirichlet b.c. to the Eigenvalue-Problem

If you use a FEM (on the variational formulation), you can discretize some continuous eigenvalue problem, $$L u = \lambda u \ \ \text{on} \ \Omega,$$ into some discrete, generalized eigenvalue problem,...
141 views

140 views

### Computing eigendecomposition of a Hermitian matrix that is almost unitary

I have a dense Hermitian matrix that is approximately unitary, so it has eigenvalues that are $\sim \pm1$. I would like to compute all the eigenvectors corresponding to the $+1$ eigenvalue (not ...
808 views

### Eigenvalue Decomposition of Hermitian Matrix in Scala

I'm working on helping my friend create code to perform the Overlap Dirac Operator and have come across one part that I'm not sure what to do. I need to compute the Eigenvalues and corresponding ...
74 views

### Efficient way to find eigenvalues of complex symmetric matrix with real off-diagonal elements

My goal is to find all eigenvalues (and eigenvectors) in a given range of magnitudes of a complex symmetric matrix with real off-diagonal elements (only diagonal elements are complex). Currently I'm ...
76 views

### Improving convergence of Jacobi iteration to Schur form

I'm using SIMD processor arrays to compute the eigen-decomposition for large numbers of small (up to $32\times 32$) matrices. For assorted technical reasons, Jacobi iteration maps well to the SIMD ...
152 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 ...
456 views

### Numerically find the nearest positive semi definite matrix to a symmetric matrix

I have a symmetric matrix $M$ which I want to numerically project onto the positive semi definite cone. To do so, I decompose it into $M = QDQ^T$ and transform all negative eigenvalues to zero. (...
247 views

### What is the most efficient way to obtain the max eigenvalue of a specific symmetric matrix via Eigen C++

Suppose I have a symmetric matrix $A_{1000\times 1000}$, which can be represented by: $A = J G J^T$ where $J$ in 1000x3 is full column rank dense matrix; $G$ in 3x3 is a nonsingular dense matrix. ...
628 views

4k views

### Eigen - Max and minimum eigenvalues of a sparse matrix

I am working with an Eigen::SparseMatrix matrix of type double. I would like to find the largest and the smallest eigenvalues. A solution of the problem is to convert it to dense and then find its ...
1k views

### Stabilizing a 3x3 real symmetric matrix eigenvalue calculation

I have many 3x3 real symmetric matrices for which I need to determine the eigenvalues. Wikipedia gives a nice non-iterative algorithm for this case, which I have translated into C++: ...
147 views

### Implementation of a $O(n \log(n))$ method to compute eigenvalues of real symmetric tridiagonal matrices

I just came upon this paper, which details the implementation of a fast method to get eigenvalues of tridiagonal symmetric matrices : Coakley, Ed S.; Rokhlin, Vladimir, A fast divide-and-conquer ...
151 views

### what do zero real parts of eigenvalues mean? Any good references?

I am solving a 1D advection problem of the the form $$dQ/dt=[A]Q$$ where {Q} is the vector of unknowns and [A] is the matrix of coefficients of spatial discretisation. I have worked out the ...
188 views

### Eigenvectors of Black-box matrix

$\DeclareMathOperator{\diag}{diag}$ Consider the generalized eigenproblem $A\mathbf{x}=\lambda B\mathbf{x}$. When solving PDEs numerically (specifically, I am interested on finding the Dirichlet ...
89 views

366 views

### Known issues with eigenvalue numerics?

Are there any known issues (such as precision issues) with $\mathsf{MATLAB}$ eig and charpoly functions for large enough $\{-1,0,+1\}$ matrices? Even if I change $1$ or $2$ entries between matrices ...
613 views

### Eigenvalues and Timestep restriction

For an equation of the kind $u_t=Au$, time step is usually defined by ensuring that the eigenvalues are within the stability region of the time-stepping scheme employed. If the eigenvalues are on ...
1k views

### Fast way to compute all eigenvalues of a dense Hermitian matrix

I am finding the eigenvalues of dense NxN Hermitian matrix which is calculated from a density operator in quantum physics. All the eigenvalues are needed as I need to calculate the sum of the absolute ...
642 views

### Generalized eigenvalue problem using ARPACK

Is it possible to solve the eigenvalue problem: $$Ax = \lambda Mx$$ using ARPACK when $A$ and $M$ are both non-symmetric complex matrices? According to this documentation, the function ...
153 views

### What is the worst case complexity of the symmetric tridiagonal QR eigenvalue algorithm?

Ignoring eigenvectors, the shifted QR algorithm for computing eigenvalues in the symmetric tridiagional case costs $O(n)$ per iteration, converges globally, and converges cubically near the end. What ...
111 views

### Analytic formula for leading eigenvector of $uu^T + vv^T$?

Let $u$ and $v$ be nonzero column vectors of size $n$ and consider the $n \times n$ positive-definite matrix $A:=uu^T + vv^T$. In this post https://math.stackexchange.com/a/112201/168758, the ...
84 views