All Questions
Tagged with matrix-decomposition eigenvalues-eigenvectors
374
questions
2
votes
1
answer
44
views
Singular value of a bidiagonal matrix?
Consider a $n\times n$ matrix:
\begin{equation}
X=\begin{bmatrix}
a &1-a & & \\
& a &1-a & \\
& & \ddots &\\
& & & 1-a\\
& & & a
\end{...
- 304
0
votes
0
answers
25
views
Best method for sequential small size Hermitian smallest eigenpair problem
I am working on a perhaps rather strange problem. To find the smallest eigenvalue and its eigenvector, for a large number (a few billions) of small (20 * 20 to 200 * 200) Hermitian matrices. These ...
- 31
0
votes
0
answers
59
views
Fast way to compute the largest eigenvector of an expensive-to-compute matrix
Consider an $N \times N$ Hermitian positive semidefinite matrix $M$. Computing the elements of $M$ is expensive so we wish to compute as few as possible. We can assume that $M$ can be approximated as ...
1
vote
1
answer
17
views
How to derive the relation about Jordan decomposition of a matrix?
Assume that $v$ is an eigenvector with an eigenvalue of $0$ in matrix $H$, and its Jordan decomposition $H^J=SHS^{-1}$ satisfies
$$
H^J=\left( \begin{matrix}
0& 1& \cdots& 0\\
0& ...
- 194
0
votes
0
answers
59
views
QR algorithm fails to converge (bad shift?)
Problem: My code-based implementation of the implicit QR algorithm fails to converge for certain special cases, and it's because those cases have bad shift values.
What are those special cases: While ...
- 433
1
vote
2
answers
70
views
What is a complete geometric interpretation of the eigendecomposition of matrices?
For reference, eigendecomposition of a matrix $A$ $\in R^{n \times n}$ is defined as:
$A = P \Lambda P^{-1}$
where $P$ is a matrix whose columns are the eigenvectors of $A$, and $\Lambda$ is a ...
1
vote
0
answers
49
views
Estimating the parameters of an ellipse
Problem definition
Consider a dataset composed by $m$ bivariate measurements
\begin{equation*}
y_j \sim \mathcal{U}(\mathcal{E}(\ell_1,\ell_2,\theta)) \qquad j=1,2,\dots,m
\end{equation*}
uniformly ...
- 729
0
votes
0
answers
37
views
Reasons of computing smallest eigenvalue $R^TR$ instead of singular value
I have problems in understanding why author of this article uses smallest eigenvalue of a cross product matrix instead of a data matrix. I know that $SVD(AA^T)=UD^2U^T$, but I don't know why not ...
- 1
1
vote
1
answer
129
views
QR factorization and eigenvectors
I really can't find an answer to this. I am writing a C code to solve the eigenvalues problem for a real symmetric matrix A. The procedure is the following.
I have a real symmetric matrix A. I put it ...
- 11
1
vote
0
answers
39
views
Eigenvalue-like problem for two matrices with coupled blocks
For given two real-symmetric matrices $A$ and $B$, both are of size $M \times M$, I'm trying to find two matrices $C_1$ (size $M\times N_1$) and $C_2$ (size $M\times N_2$) such that
$$
AC_1 = C_1\...
- 11
0
votes
1
answer
74
views
The probability value of the three matrices
Determine the probability value that the following three matrices have real eigenvalues. For example, if the random variables $A$ and $B$ are given by a uniform distribution $(0,1)$, with $A$ and $B$ ...
0
votes
0
answers
35
views
How to calculate the square root of a special matrix
I have met a question to calculate the square root of a matrix in $\mathbb{R}^{s+1}$.
$$A = \left(\begin{array}{cc}
\sigma_{11}^{2} & \sigma_{11}^{1/2}\Sigma_{1S}\Sigma_{SS}^{1/2}\\
\sigma_{11}^{1/...
- 1
1
vote
0
answers
29
views
Name for 3 full diagonal matrix
I was wondering if there is a name or a terminology for a matrix with 3 full diagonals. It is similar with tridiagonal matrices, where the later is only centered at diagonal.
For such a matrix,
$$\...
- 337
2
votes
1
answer
156
views
When is a symmetric matrix $A ∈\Bbb R^{3×3}$ with three distinct eigenvalues equal to $A^n$?
How do you find a symmetric matrix $A ∈\Bbb R^{3×3}$, knowing that:
$A^n$ $= A$ for some $n > 1$
$A$ has three distinct eigenvalues
and we are given two (orthogonal) eigenvectors $[1, 2, 2]^T$ and ...
- 21
0
votes
0
answers
14
views
I am trying to derive the bound of the row sum of a transformed matrix
We have a symmetric binary matrix $K_n$ with diagonals all 1: $K_n = \left( K_{ij} \right)_{i,j=1}^n$ and apply eigenvalue decomposition to it:
$$
{K}_n:
= Q_n \Lambda_n Q_n^{\top}
= Q_n \max\{ \...
- 1