Tagged Questions
1
vote
1answer
47 views
Boundary-value problem in differential equations
Consider the problem:
$$u^{(4)} + \lambda u = 0, \ \ \ 0<x<\pi; \ \ \ u(0) = u(\pi) = u''(0) = u''(\pi) =0$$
Find the eigenvalues.
How should one proceed about this problem? I am complete ...
2
votes
3answers
121 views
Spectrum of eigenvalues and eigenfunctions
Our O.D.Es professor had the "amazing" idea of heavily introducing advanced linear algebra material (which is not an official prerequisite for the course) along with boundary value problems. Not being ...
1
vote
0answers
123 views
Would this be bounded?
Assume that $M$ is an $m$ by $m$ ($m$ is an even number) symmetric
positive-semi-definite matrix with exactly $m/2$ positive eigenvalues
and every entry of $M$ is less than $1$. Let $I_{r}$ be an $m$
...
1
vote
1answer
53 views
Proof that these Hessian matrix identities are similar matrices
I am wondering if $Q, P$ are similar matrices where for a function
$f:\mathbb{R^n}\to\mathbb{R}$ and for a diagonal matrix $D$
$Q=I-D^{-1}\nabla^2f(x)$ and $P=I-D^{-1/2}\nabla^2f(x)D^{-1/2}$.
...
5
votes
1answer
294 views
Distinct eigenvalues for integral operator?
Is there some sufficient condition on the kernel $K$ of a (say, finite rank, to simplify) integral operator
$$
\mathcal K:f(x)\in L^2(\mathbb R)\mapsto \int K(x,y)f(y)dy
$$
so that it has all its ...
3
votes
0answers
114 views
Does matrix convergence in $L^p$ imply convergence of the eigenvalues in $L^p$?
Let $A_n(x)$ be a sequence of symmetric matrix functions that converges in $L^p(\Omega)$ to $A(x)$. Is it true that the eigenvalues of $A_n(x)$, or a subsequence of these, converge to the eigenvalues ...
