All Questions
Tagged with eigenvector ca.classical-analysis-and-odes
8 questions
3
votes
0
answers
47
views
What is the supremum of 1-dim Hausdorff measure of the nodal set of Neumann eigenfunction $u$ for planar convex domain
All descriptions of this question are limited to 2-dimension for simplicity. Recently, I read some papers on the nodal set of Laplacian eigenfunctions. Denote $\Sigma=\{u(x)=0\}$ be the nodal set of a ...
- 131
2
votes
1
answer
1k
views
Diagonalizing a symmetric block matrix
Let us consider the matrix
$$ A = \begin{pmatrix} a & c+ib \\ c-ib& a \end{pmatrix},$$
then this matrix has eigenvalues $a\pm \sqrt{c^2+b^2}.$
Now, let us consider a block matrix
$$ A = \begin{...
- 29
4
votes
2
answers
372
views
How close to uniform are Perron-Frobenius eigenvectors?
Let $A=(a_{i,j})$ be a square matrix with non-negative entries. (Assume $A$ is symmetric, if it helps.) Let $v$ be a Perron-Frobenius eigenvector. What do we need to assume about $A$ in order to have ...
- 20.2k
-2
votes
1
answer
713
views
The spherical harmonics are the EIGENVECTORS of Beltrami operator [closed]
In the well-known book "THE PRINCETON COMPANION TO MATHEMATICS" page 296, it is indicated that the spherical harmonics are the EIGENVECTORS of the Beltrami operator. In the document Spectral Geometry ...
- 123
1
vote
0
answers
122
views
Algorithm for finding eigenfunctions
I have an $ L^2(\mathbb{R}) $ operator that looks like
$$
\Omega = \int \partial\phi(a, b)\ \ |b, a\rangle \langle b, a |,
$$
where $ \langle x | a, b \rangle = f_a(x - b) e^{x^2/2} $ and $ f_a \in L^...
user66731
0
votes
1
answer
30k
views
Difference between Principal Component Analysis(PCA) and Singular Value Decomposition(SVD)? [closed]
I am confused between PCA and SVD.
The wikipedia page for PCA has this line. "PCA can be done by eigenvalue decomposition of a data covariance matrix or singular value decomposition of a data matrix, ...
- 141
17
votes
5
answers
2k
views
Can always a family of symmetric real matrices depending smoothly on a real parameter be diagonalized by smooth similarity transformations?
This question is related to another question, but it is definitely not the same.
Is it always possible to diagonalize (at least locally around each point) a family of symmetric real matrices $A(t)$ ...
- 4,312
20
votes
2
answers
8k
views
Conditions for smooth dependence of the eigenvalues and eigenvectors of a matrix on a set of parameters
Let $A\in\mathcal M_n$ be an $n\times n$ real [symmetric] matrix which depends smoothly on a [finite] set of parameters, $A=A(\xi_1,\ldots,\xi_k)$. We can view it as a smooth function $A:\mathbb R^k\...
- 4,312