# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 $m$ ...
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### 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 $P=I-D^{-1/2}\nabla^2f(x)D^{-1/2}$. ...
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 ...
### 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 ...