# Questions tagged [eigensystem]

An eigenvector of an operator is a vector such that the action of the operator is the same as multiplication by a constant, called the eigenvalue. The eigensystem of an operator is the set of all such eigenvectors and their associated eigenvalues.

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### Eigenvector with maximum overlap

Given a matrix $M$ and a vector $v$, is there an efficient method to find the normalized eigenvector of $M$ that is closest to $v$, in that it has maximal overlap. More explicitly, a vector $v$ can be ...
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### Approximate spectrum of a large matrix

I want to compute the spectrum (all the eigenvalues) of a large sparse matrix (hundreds of thousands of rows). This is hard. I am willing to settle for an approximation. Are there approximation ...
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### Large-scale generalized eigenvalue problem with low rank LHS matrix

Assume that we have generalized eigenvalue problem: $B^HB\textbf{x} = \lambda A\textbf{x}$ where $A$ is an nxn Hermitian sparse matrix (n is very large, so we do not have $A^{-1}$ but can solve ...
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### Sparse hermitian eigensystems: are there better techniques than Arpack or TRLan?

As a part of other work I need to solve relatively large (~1E5x1E5) and sparse (~100 non-zero elements in each raw in few blocks) hermitian eigensystems. Usually only few eigenvalues+vectors are ...
201 views

### diagonalization of matrix - omitting small matrix elements

I was wondering whether there is some theorem that allows me to put an upper bound on the error introduced by omitting small matrix elements from a matrix before diagonalization. Let's assume we ...
Let $A\in \mathbb{R}^{n\times n}$ symmetric and positive semidefinite, and $\omega\in \mathbb{R}\setminus\{0\}$. I am interested in solving the following linear system for a range of values of $\... 0answers 126 views ### What strategies / decompositions would be useful to solve the following linear system repeatedly if I only care about time to solution? Let$A\in \mathbb{R}^{n\times n}$be symmetric positive semidefinite, and$B\in \mathbb{R}^{n\times n}$be symmetric positive definite. Suppose$B$is block diagonal so it is easy to invert. (We ... 1answer 963 views ### Spectral decomposition with eigenvalue shift Suppose a square, real and symmetric matrix$G\in\mathbb{R}^{n\times n}$is given, and it is known to have one zero eigenvalue associated with all ones eigenvector,$1_n$. I'm aware that the (possibly)... 4answers 2k views ### Finding the square root of a Laplacian matrix Suppose the following matrix$A$is given $$\left[\begin{array}{ccc} 0.500 & -0.333 & -0.167\\ -0.500 & 0.667 & -0.167\\ -0.500 & -0.333 & 0.833\end{array}\right]$$ with ... 1answer 1k views ### generalized eigenvalue problem I need to solve a real generalized eigenvalue problem$Ax= \lambda Bx(*)$A and B are calculated from equations below: $$A=\sum_{i,j=1}^{N}W_{ij}(K_{i}-K_{j})\beta\beta^{T}(K_{i}-K_{j})^{T}$$ $$B=\... 2answers 293 views ### Is it possible to ignore/discard part of a matrix when finding eigenvalues? I have have multiple large matrices for which I need to find the largest absolute eigenvalue. I know that there is a large submatrix that does not vary. Is it possible to ignore/discard the submatrix? ... 2answers 3k views ### Fast algorithms to find the eigenvalues of some matrix on intervals of interest I am curious how to quickly compute the eigenvalues for arbitrary matrices, sparse or dense, restricted on some given interval of interest. Suppose we have an arbitrary n\times n matrix A, ... 3answers 6k views ### Algorithm for Principal Eigenvector of a Real Symmetric 3x3 Matrix I have a 3x3 covariance matrix (so, real, symmetric, dense, 3x3), I would like it's principal eigenvector, and speed is a concern. Is there a fast algorithm for this specific problem? I've seen ... 2answers 198 views ### Finding dominant eigenvectors of an operator that is small but costly to evaluate Suppose I have a symmetric linear operator A:\mathbb{R}^k \rightarrow \mathbb{R}^k where k is "small" (eg., k=100), and I want to find it's first few eigenvectors, (eg., 10 eigenvectors). If ... 0answers 415 views ### Iteratively finding both left and right eigenvectors for non-symmetric complex matrix I have a complex, non-Hermitian matrix \mathbf{A}, for which I need to find a few eigenvalues and eigenvectors in the generalised eigenvalue problem:$$\mathbf{A}\cdot \mathbf{x} = \lambda \mathbf{... 0answers 47 views ### When numerically computing eigenstates during a coupled-mode-space NEGF calculation, do phases matter? The coupled mode space NEGF method for computing transistor characteristics involves expanding the electronic wavefunction in a mode space basis $$\Psi(x,y,z) = \sum_n\phi_n(x)\xi_n(y,z;x)$$ where$\...
I wrote a simple script to generate random polynoimals $\displaystyle f(z)= \sum_{k=0}^N a_k \frac{z^k}{\sqrt{k!}}$ of high degree and find their roots. For more discussion on random polyomials see ...