7
votes
1answer
111 views

Fast algorithm for approximating Eigenvalue distribution of large sparse matrix

I am interested in the eigenvalue distribution of a huge $2^{16}$x$2^{16}$ Hermitian sparse matrix with spectrum contained in $[-1,1]$. That is I don't need to know all eigenvalues exactly, but rather ...
3
votes
0answers
64 views

How to quickly approximate the eigenvectors of a symmetric matrix

Given a symmetric $n \times n$ matrix $A$, is there any algorithm that can quickly approximate all of its eigenvectors? By "quickly", I mean with time complexity less than $\mathcal{O}(n^3)$.
1
vote
1answer
127 views

Computing the number of positive and negative eigenvalues

Given a $n \times n$ symmetric matrix $A$ with integers as entries I would like to compute the number of strictly negative $\rm{nn}(A)$ and positive $\rm{np}(A)$ eigenvalues of $A.$ My question is ...
-1
votes
1answer
154 views

Algorithm of extract eigenvalue and eigenvector from matrix using lower triangular matrix

I want to compute the eigenvalue of a matrix with this transform: $$\text{BaseMatrix}\rightarrow \text{LowerTrangularMatrix}$$ and my algorithm of this transform is this: ...
1
vote
1answer
165 views

Determine direction of eigenvector

Suppose that $A \in \mathbb{R}^{n \times n}$ and $B \in \mathbb{R}^{n \times m}$ are integer matrices. Let $P$ be the unbounded polytope in $\mathbb{R}^n$ given by $$B \cdot x \geq 0$$ As there is no ...
3
votes
2answers
183 views

Is the QR algorithm for computing eigenvalues efficient for today's standards?

I was looking at the QR factorization algorithm of a matrix to approach eigenvalues. At the Wikipedia page they state that it was developed in the 50's and took over the LR algorithm. They also state ...
4
votes
1answer
373 views

Characterizing a real symmetric matrix $A$ as $A = XX^T - YY^T$

In my personal research and quest to better understand the subject, I have noticed something concerning the Cholesky factorization of symmetric matrices. Everything I have read states that a symmetric ...
1
vote
0answers
139 views

The fastest algorithm of computing Principal eigenvector of a non-negative-entries matrix

I am studying the QR algorithm, is it the fastest one in this situation?