# Tagged Questions

182 views

### Eigenvalues gone wild

I added some significant details to this problem, as it was apparently not clear to everyone what I want to know: This is a question about convergence of eigenvalues which essentially came up in ...
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### proving that $\max Q(x)=\lambda_\max$

Let $Q(x)$ be quadratic form. Prove that $\max_{\|x\|=1}Q(x)=\lambda_\max$. $Q$ is symmetric so it can be presented as $$\langle Ax,x\rangle$$ where $A$ is matrix which on its diagonal appears ...
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### Prove that an eigenvector is the maximum of a symmetric matrix

Let $f : S^{n-1} \rightarrow \mathbb{R}, x \mapsto x^TAx$ ( A is a symmetric matrix), then an eigenvector $\xi$ of A is a local maximum of this function. We are supposed to prove this in 6 steps and ...
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### Given $\mathbf{A}$ stable (all negative eigenvalues), produce a bound on $\|\mathbf{B}\|$ such that…

Given a system: $\dot{\mathbf{x}}=\left(\mathbf{A}+\mathbf{B}\right)\mathbf{x}$ Can you bound $\|\mathbf{B}\|$ s.t. the origin of $\mathbf{x}$ is exponentially stable using a Lyapunov function? ...
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### Distinguishing between the different eigenvalues

Consider the symmetric matrix $$A=\begin{pmatrix} 2 & t & \cos t-1 \\ t & 2 & 0 \\ \cos t-1 & 0 & 2 \end{pmatrix}.$$ The (real) eigenvalues of $A$ can be found easily using ...
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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 ...