# Tagged Questions

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### Determinant Formula for Tri-Diagonal Matrix

for an assignment in numerical analysis, I need to find the eigenvalues of a matrix with values only in the diagonal, upper diagonal and lower diagonal. I guess there is an easy formula for this sort ...
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### Complexity of the power method

I'd like to find out what the complexity of the power method is depending on the size of the matrix $A \in \mathbb{R}^{n\times n}$ given that the algorithm runs until a certain stop criterion. I.e. ...
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### Numerical Computation for K smallest eigenvalues of a large Real Symmetric Matrix with restricted methods

I'm writing some code on a distributed platform, using some programming language like Hadoop, and now I need to calculate the K smallest eigenvalues for a Large Matrix. K is a small constant at most ...
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### Power iteration

If $A$ is a matrix you can calculate its largest eigenvalue $\lambda_1$. What are the exact conditions under which the power iteration converges? Power iteration Especially, I often see that we ...
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### Influence of preconditioning on degenerate eigenvectors

I'm using a hierarchical decomposition of a sparse matrix $A$ as suggested here. I find that the method essentially finds eigenvectors using the QR algorithm. $A$ has some eigenvalues with ...
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### How to compute the eigenvalue condition number of a matrix

How to compute the eigenvalue condition number, $\kappa(4,A)$, of a matrix $A$ $$A = \begin{bmatrix} 4 & 0 \\ 1000 & 2\end{bmatrix}$$ I am a bit stuck on how to proceed solving this problem ...
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### Two Lagrange multipliers with one equation

I have an equation as below, $$Rw = \lambda_1R_aw + \lambda_2R_bw$$ where, $R$, $R_a$, and $R_b$ are positive definite at least semi-positive definite and Hermitian matrix. $\lambda_1$ and ...
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### Math hack for solving system of equations

Is it a "standard" Math/Numerical-Analysis hack to add a relatively small number e.g. 1*10E-5 to the diagonal of a squared matrix to ensure LU Decomposition (or whichever decomposition algorithm is ...
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### Power iteration smallest eigenvalue?

I need to write a program which computes the largest and the smallest (in terms of absolute value) eigenvalues using both Power Iteration and Inverse Iteration. I can find them using the Inverse ...
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### Sturm-Liouville problem: eigenvalues

I have a Sturm-Liouville problem $$y'' + \lambda^2 y = 0, \\ y'(0) + \alpha_1 y(0) = 0, \\ y'(L) + \alpha_2 y(L) = 0,$$ where $\alpha_1 \alpha_2 \neq 0$. I found that eigenvalues are ...
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### Numerical Computation of Eigenvalues

I am trying to find the first few eigenvalues of an operator defined by the following PDE:  \begin{cases} -\Delta u +(1-\varphi)u=\lambda u, & \text{ on }\Omega = [0,1]^2 \\ u=0 & \text{ ...
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### How to determine the N-smallest eigenvalues of a symmetric matrix using the Power Method?

I was assigned to make a program that finds the largest, the N-largest, the smallest and the N-smallest eigenvalues of a symmetric matrix, using the Power Method. So far, I've been able to succesfully ...
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### variant eigenvector problem

I have the following problems when solving a linear equation. Let $A=(a_{i,j})_{n \times n}$ be a non-negative matrix with $a_{i,j} \in (0,1)$, and let $0<r<1$ be a scalar. Now we define a ...
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### Is there a version of the Gershgorin circle theorem that is suitable for nearly triangular matricies?

The Gershgorin circle theorem, http://en.wikipedia.org/wiki/Gershgorin_circle_theorem, gives bounds on the eigenvalues of a square matrix, and works well for nearly diagonal matrices. For a ...
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### program for eigenvalue calculation

I have a n x n matrix. I would like to (a) take successively higher powers of the matrix and then multiply by projection vectors until the resulting vectors differ by only a scalar factor. (b) ...
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### Two linearly independent eigenvectors with eigenvalue zero

What is the only $2\times 2$ matrix that only has eigenvalue zero but does have two linearly independent eigenvectors? I know there is only one such matrix, but I'm not sure how to find it.
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### Show the space spanned is an invariant subspace

Let $A$ be real and let $\lambda = \alpha + i \beta$ be a complex eigenvalue of $A$ with eigenvector $x + iy$, show that the space spanned by $x$ and $y$ is an invariant subspace of $A$. What I ...
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### Using the Arnoldi Iteration to find the k largest eigenvalues of a matrix

I'm trying to obtain a general understanding of this algorithm which determines the k-largest eigenvalues of a matrix $A\in \mathbb{R}^{n\times n}$. How I see it: power iteration: take random ...
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### Library for Jacobi eigenvalue algorithm [closed]

I am looking for a C or C++ or fortran library that implements the Jacobi eigenvalue algorithm: http://en.wikipedia.org/wiki/Jacobi_eigenvalue_algorithm do you know if it is available?
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### Shifted Power Method

Using the shifted power method I find the eigenvalue (of the matrix A) farthest from a number $\mu$ and the corresponding eigenvector . In the method I follow the below steps: I first compute the ...
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### What is a robust and reliable way/library for eigenvalues of 3x3 matrices?

I use Eigen to compute the eigenvalues of symmetric matrices. The problem is, that sometimes the matrices not nice at all numerically. Because of this, I get NaN among the eigenvalues. I have tested ...