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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How is the SVD of a matrix computed in practice
How does MATLAB, for instance, calculate the SVD of a given matrix? I assume the answer probably involves computing the eigenvectors and values of A*A'. If that is ...
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1answer
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Ground state eigenvector different for different eigen solvers (differs by negative sign in the elements). Does it matter?
Here is some code that hopefully clearly illustrates what I'm doing:
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3answers
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computing the determinant of a dense nonsymmetric 100x100 matrix having very big and very small eigenvalues
The motivation for my question is the following: in one of Project Euler questions there is a need to count the spanning trees of a rectangular grid graph of dimension 100x500. By the Matrix-Tree ...
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What is the fastest way to calculate the largest eigenvalue of a general matrix?
EDIT: I am testing if any eigenvalues have a magnitude of one or greater.
I need to find the largest absolute eigenvalue of a large sparse, non-symmetric matrix.
I have been using R's eigen() ...
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Specialized methods for symmetric tridiagonal generalized eigenvalue problems
I have to solve generalized eigenvalue problems $Ax = \lambda Bx$ where $A$ and $B$ are both tridiagonal, $B$ is symmetric positive definite and real, but $A$ is only complex symmetric (not definite ...
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1answer
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Numerical Methods for minimizing a Non-Differentiable Convex Function of Several Variables
I have a multi-variable convex continuous function which is not differentiable. I am interested to know about different numerical techniques, possibly also references to them, used for this.
Read ...
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2answers
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Finding the distribution (histogram) of eigenvalues for large sparse matrices
Are there any existing programs that are able to compute the (approximate) distribution of eigenvalues for very large (symmetric) sparse matrices?
Note that I do not need the eigenvalues themselves, ...
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1answer
76 views
Simple Lanczos algorithm code to obtain eigenvalues and eigenvectors of a symmetric matrix
I would like to write a simple program (in C) using Lanczos algorithm. I came across a Matlab example which helped me to understand a bit further the algorithm, however from this piece of code I can't ...
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1answer
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On implementations of the complex QR decomposition
I have an mathematical theorem on the QR decomposition, which relies on the QR decomposition of an invertible square complex matrix always constructing a triangular matrix with real diagonal entries.
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2answers
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Implementing PageRank using the Power Method
I am trying to implement the PageRank algorithm described in this paper (Fig. 1). Here is the breakdown of the steps:
where:
pT is a probability distribution ...
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1answer
235 views
Power Iteration on general matrices (with higher multiplicity of dominant eigenvalue)
To compute the eigenvector corresponding to a dominant eigenvalue of a matrix $A\in\mathbb{R}^{n\times n}$, one could apply the Power Iteration: $$v_1=\frac{Av_1}{\|Av_1\|}.$$
1) in case $A$ is ...
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2answers
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An eigenvalue algorithm to solve constrained quadratic form minimization
I have a quadratic form $\mathbf{x}^T A \mathbf{x}$ (where $A\in \mathbb{R}^{n\times n}$ is symmetric matrix and $\mathbf{x}\in \mathbb{R}^n$) that I want to minimize given the normalization ...
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Manipulating a generalized eigenvector problem to plain eigenvector problem
Let $X\in\mathbb{R}^{n\times p}$ denote a matrix with $p$ linearly-independent columns, and let $L\in\mathbb{R}^{n\times n}$ denote a symmetric matrix. Furthermore, let $D\in\mathbb{R}^{n\times n}$ ...
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Eigenvectors with the Power Iteration
To compute the eigenvector corresponding to dominant eigenvalue of a symmetric matrix $A\in\mathbb{R}^{n\times n}$, one used Power Iteration, i.e., given some random initialization, ...
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Fastest way to find eigenpairs of a small nonsymmetric matrix on a GPU in shared memory
I have a problem where I need to find all positive (as in the eigenvalue is positive) eigenpairs of a small (usually smaller than 60x60) nonsymmetric matrix. I can stop calculating when the eigenvalue ...
