All Questions
16 questions with no upvoted or accepted answers
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
4
votes
0
answers
447
views
How to find eigenvalues of following block matrices?
Is there a procedure to find the eigenvalues of A?
$$A=\begin{bmatrix}X & I &&&&&&&&& 0\\I & 0 & P &&&&&&&&\\& P^t ...
- 41
4
votes
0
answers
2k
views
What is the time complexity of the largest singular value and its vectors?
Full zero-error SVD on an $m \times n$ matrix $A$ would cost $O(\min(m^2n,mn^2))$. What is the time complexity if we need only the largest singular value and its corresponding vectors? I think it is $...
- 123
4
votes
0
answers
84
views
Matrices with almost constant coefficient have a simple eigenvalue
As a by-product of a general result for bounded operators of a Banach space, I have the following:
A matrix $L=(\ell_{ij})_{ij}$ that has almost constant coefficients in the sense that for some $c$,...
- 14.4k
3
votes
0
answers
220
views
Eigenvalues and eigenvectors of nonsymmetric complex tridiagonal matrix
I wonder if it is possible to find analytically all eigenvalues and eigenvectors of the following $2n \times 2n$ non-symmetric complex tridiagonal matrix
$$M = i \begin{pmatrix}
0 & a & 0 &...
2
votes
0
answers
86
views
Smallest eigenvalue of certain PD matrix decreases under sparse perturbation
Let $\omega_1<\dots<\omega_n\in\mathbb{R}$. Then, define $G\in\mathbb{C}^{n\times n}$ such that $G_{k\ell}=\frac{1}{1-i(\omega_\ell-\omega_k)}$. For example, if $n=3$ we obtain $$ G=\begin{...
- 83
1
vote
0
answers
179
views
QR algorithm for eigenvalues and eigenvectors of large symmetric matrices
I am trying to write a QR algorithm in Python for eigenvectors and eigenvalues finding for large symmetric matrices,
My initial thought was to use Householder transformation with a Wilkinson shift ...
1
vote
0
answers
293
views
Eigenvalue decomposition of normalized adjacency matrix
Let $A$ be an adjacency matrix of undirected graph $G$, where $G$ is a connected graph. The normalized adjacency matrix is defined as $\hat{A}=D^{-1/2}AD^{-1/2}$, where $D$ is degree matrix of graph $...
- 11
1
vote
0
answers
86
views
Solution of a manipulated equation vs the maximum eigenvalue and eigenvector of a non-negative matrix
Lets assume we have the following equation:
$AU=\lambda U \Rightarrow\left[
\begin{array}{c|c|c}
0 &A_{12}&A_{13}\\
\hline
A_{21}& 0& A_{23}\\
\hline
A_{31}&A_{32}&0
\end{...
- 21
1
vote
0
answers
92
views
An inequality concerning the eigenvalues and eigenvectors of an SPD matrix
Let $Ax_i=\lambda_ix_i, \ (i=1,\cdots,n)$ be an eigensystem of the symmetric positive-definite diagonally-dominant matrix $A=\{a_{ij}\}$. Let
$$b_{jk}=\sum_{i=1}^{n}{\frac{(x_i(j)-x_i(k))^2}{\...
- 11
1
vote
0
answers
171
views
Eigenvalues of non-negative block matrices
$B$ is a non-negative irreducible block matrix as follows:
$$B=
\left[
\begin{array}{c|c|c}
0 &B_{12}&B_{13}\\
\hline
B_{21}& 0& B_{23}\\
\hline
B_{31}& B_{32}&0
\end{array}
\...
- 21
1
vote
0
answers
270
views
Eigenvalue of product of self adjoint compact operators
Suppose A is a self adjoint $m \times m$ real matrix with eigenpairs $\{e_j, \lambda_j\}$ such that $\lambda_j > \lambda_{j + 1}$. Let $B$ be another self adjoint real $m \times m$ matrix such that ...
- 157
0
votes
0
answers
149
views
Diagonalizing a specific case of symmetric block matrix
Let's consider the following block matrix
$$ M = \begin{pmatrix}D&A^T\\A&-D\end{pmatrix},$$
where $A$ and $D$ are $n \times n$ matrices. The diagonal matrix $D$ is defined by $D_{kk} = k \...
- 1
0
votes
0
answers
166
views
Minimize a vector from a matrix operation
I want to minimize a certain vector that results from a matrix operation with some constraints and i don't exactly know how to tackle this problem.
Lets say we have
$$
(L+A)*s = v
$$
L is the ...
- 101
0
votes
0
answers
79
views
Eigendecomposition of $A=I+BDB^H$
Suppose that we have $$A = I_m + BDB^H$$ where matrix $A$ is $m \times m$, matrix $B$ is $m \times k$, $BB^H \neq I_m$ and $D$ is a $k \times k$ diagonal matrix. Can we obtain the eigendecomposition ...
- 13
-3
votes
1
answer
270
views
Relationship of eigenvalue/eigenvector of hermitian matrix R and QRQ (Q is diagonal)
For a hermitian matrix R and a diagonal one Q, is there any relationship between eigenvalues/eigenvectors of R and QRQ?
To be specific, assuming the eigenvalue decomposition of R is R=VDV*, then can ...
- 101