All Questions
16 questions with no upvoted or accepted answers
7
votes
0
answers
384
views
Concept of eigenvector restricted to nonnegative entries
Let $X\in \mathbb{R}^{n\times n}$ be a positive semidefinite matrix. The leading eigenvector $v\in \mathbb{R}^n$ of $X$ is the solution to the problem
$\arg \max_{v:\lVert v\rVert_2=1} \lambda\quad$ ...
- 373
6
votes
0
answers
96
views
Finding the maximal component of a vector in sublinear time
Given a vector $u \in \Bbb R^n$, finding the value of the largest component of $u$ needs linear time in $n$. However, what if we additionally know that $u$ lies in some linear subspace $U \subset \Bbb ...
- 13.6k
5
votes
0
answers
208
views
Perturbation of Neumann Laplacian
Consider the $N \times N$ matrix
$$A_{\alpha}=\begin{pmatrix} \lambda_1 & -1 & -\alpha & 0 & \cdots & 0\\
-1 & \lambda_2 & -1 & -\alpha & \cdots & 0\\
-\alpha &...
- 73
5
votes
0
answers
392
views
Preconditioner for finding the smallest eigenpairs of a large, but structured, matrix
I'm trying to find the eigenvector corresponding to the second smallest eigenvalue of a large $(4,000,000 \times 4,000,000)$ matrix $L$. $L$ is a graph Laplacian, with the following structure: $L = D -...
- 500
4
votes
0
answers
447
views
How to find eigenvalues of following block matrices?
Is there a procedure to find the eigenvalues of A?
$$A=\begin{bmatrix}X & I &&&&&&&&& 0\\I & 0 & P &&&&&&&&\\& P^t ...
- 41
4
votes
0
answers
2k
views
What is the time complexity of the largest singular value and its vectors?
Full zero-error SVD on an $m \times n$ matrix $A$ would cost $O(\min(m^2n,mn^2))$. What is the time complexity if we need only the largest singular value and its corresponding vectors? I think it is $...
- 123
3
votes
0
answers
220
views
Eigenvalues and eigenvectors of nonsymmetric complex tridiagonal matrix
I wonder if it is possible to find analytically all eigenvalues and eigenvectors of the following $2n \times 2n$ non-symmetric complex tridiagonal matrix
$$M = i \begin{pmatrix}
0 & a & 0 &...
2
votes
0
answers
79
views
Conditions on a $n\times n$ Hermitian matrix such that its extremal eigenvectors have equal magnitude entries
Is it possible to find (necessary and sufficient) conditions on a general $n\times n$ Hermitian matrix $A$, such that its extremal eigenvectors (the eigenvectors corresponding to the maximum and ...
- 31
2
votes
0
answers
28
views
Comparison of principal diagonals of two positive definite matrix
Let us consider a matrix with positive elements: ${\bf X}^{k\times2}=[X_1:\ldots:X_k]'$ with $X_i=(1,X_{1i})',\;i=1,\ldots,k$. Also consider ${\bf X}^{(-1)}$ as the Moore-Penrose inverse of ${\bf X}$ ...
- 213
1
vote
0
answers
179
views
QR algorithm for eigenvalues and eigenvectors of large symmetric matrices
I am trying to write a QR algorithm in Python for eigenvectors and eigenvalues finding for large symmetric matrices,
My initial thought was to use Householder transformation with a Wilkinson shift ...
1
vote
0
answers
86
views
Solution of a manipulated equation vs the maximum eigenvalue and eigenvector of a non-negative matrix
Lets assume we have the following equation:
$AU=\lambda U \Rightarrow\left[
\begin{array}{c|c|c}
0 &A_{12}&A_{13}\\
\hline
A_{21}& 0& A_{23}\\
\hline
A_{31}&A_{32}&0
\end{...
- 21
1
vote
0
answers
171
views
Eigenvalues of non-negative block matrices
$B$ is a non-negative irreducible block matrix as follows:
$$B=
\left[
\begin{array}{c|c|c}
0 &B_{12}&B_{13}\\
\hline
B_{21}& 0& B_{23}\\
\hline
B_{31}& B_{32}&0
\end{array}
\...
- 21
1
vote
0
answers
270
views
Eigenvalue of product of self adjoint compact operators
Suppose A is a self adjoint $m \times m$ real matrix with eigenpairs $\{e_j, \lambda_j\}$ such that $\lambda_j > \lambda_{j + 1}$. Let $B$ be another self adjoint real $m \times m$ matrix such that ...
- 157
0
votes
0
answers
103
views
Eigenvectors of tridiagonal hermitian matrix
In my paper, I investigate the coordinates of the eigenvectors of a hollow tridiagonal hermitian matrix, which is defined as:
\begin{align*}
Q_n =
\begin{pmatrix}
0 & q_{1,2} & 0 & 0 & ...
- 1
0
votes
0
answers
149
views
Diagonalizing a specific case of symmetric block matrix
Let's consider the following block matrix
$$ M = \begin{pmatrix}D&A^T\\A&-D\end{pmatrix},$$
where $A$ and $D$ are $n \times n$ matrices. The diagonal matrix $D$ is defined by $D_{kk} = k \...
- 1
0
votes
0
answers
236
views
Eigenvectors of a matrix
Let $M$ be a square matrix of order $n\times d$. Let $\xi_{1},\dots,\xi_{n\times d}$ be an orthonormal basis of $\mathbb{R}^{n\times d}$ formed of eigenvectors of $M$. We have
$$\xi_{i}=(\lambda_1, 0,...
