0
votes
0answers
7 views

Stieltjes transformation of e.d.f. sample eigenvalues

If the eigen-decomposition of sample covariance matrix is $S=PDP'$ where $D$ is a diagonal matrix with eigen value of $S$ and $P$ are eigenvectors. If we define the empirical distribution function of ...
0
votes
1answer
33 views

Finding Eigenvalues with Gershgorin-Discs

$A=\begin{pmatrix}-5 & 0 & 0 \\ 2 & 2 & 1 \\ 3 & -5 & 4\end{pmatrix}$ Find the Gerschgorin-discs, where the eigenvalues of $A$ lie. According to the formula if we ...
2
votes
0answers
41 views

Standard symmetric tridiagonal matrix Eigenvalue decomposition algorithm?

Hi I am trying to generate an arbitrary Gauss quadrature rule by using the Golub-Welsh algorithm (here). I need to code this on C++ for my personal project. This algorithm involves the eigenvalue ...
2
votes
1answer
64 views

Quick way of finding the eigenvalues and eigenvectors of the matrix $A=\operatorname{tridiag}_n(-1,\alpha,-1)$

Matrix $A=\operatorname{tridiag}_n(-1,\alpha,-1)$ has the eigenvalue: $\lambda_i=\alpha-2\cos(i\theta),$ $i=1,\dots,n$ and the corresponding eigenvectors are: ...
1
vote
2answers
89 views

Power iteration sign of eigenvalue?

I need to write a program which computes all eigenvalues and corresponding eigenvectors. I'd like to use power iterations method (I know that it's not good but it's really necessary). my algorithm ...
0
votes
1answer
54 views

computing eigen values

find all the smallest eigenvalues of the matrix $$ A= \begin{pmatrix} 4 &3& 0\\ 3 &4& 1\\ ...
2
votes
0answers
22 views

Does eigenvalue theory remain in numerical applications?

If I have a symmetric matrix $A \in \mathbb{Q}^{n \times n}$ in matlab, then in theory it is guaranteed that $A$ has a orthogonal basis of eigenvectors and real eigenvalues. Does this remain in ...
4
votes
4answers
134 views

How to find 2x2 matrix with non zero elements and repeated eigenvalues?

I need to find a 2x2 matrix with non zero elements that has eigenvalue = 1 repeated (double). How can i do that? Thanks!
0
votes
0answers
37 views

is 'chasing the bulge' in the implicit QR algorithm exactly the same as reducing a general matrix to hessenberg form?

When performing the implicit QR algorithm, there's a part where you 'chase the bulge' down the diagonal. While it may not necessarily be numerically or computation-time equivalent, is that ...
0
votes
1answer
104 views

How to show all eigenvalues are positive?

Could you help me to show that the following matrix has all its eigenvalues positive? $$H= \begin{bmatrix} \sum_{k=1}^ng_1(x_k)^2 & \sum_{k=1}^ng_1(x_k)g_2(x_k) & \cdots & ...
0
votes
1answer
123 views

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 ...
4
votes
2answers
162 views

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 ...
3
votes
2answers
2k views

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 ...
0
votes
1answer
295 views

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 ...
3
votes
1answer
95 views

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{ ...
2
votes
2answers
388 views

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 ...
0
votes
1answer
64 views

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 ...
5
votes
2answers
289 views

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 ...
0
votes
3answers
358 views

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) ...
0
votes
3answers
526 views

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.
4
votes
1answer
122 views

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 ...
6
votes
1answer
1k views

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 ...
-1
votes
2answers
787 views

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?
1
vote
0answers
554 views

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 ...
2
votes
3answers
375 views

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 ...
3
votes
1answer
318 views

Eigenvectors of a matrix reduced to tridiagonal

I am implementing an algorithm to calculate eigenvalues and eigenvectors of a symmetric matrix in a GPU. In order to calculate the eigenvalues I first reduced the matrix to the tridiagonal form using ...
3
votes
1answer
305 views

How do iterative methods applied to the companion matrix of a polynomial $p(\lambda)$ relate to $p$ itself?

A few days ago, I had a vague question in my mind about "matrix methods" for finding roots of a polynomial. Now I can ask at least a semi-precise question, thanks to the post How to calculate complex ...