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6 votes
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Eigenvalues of splitting scheme

In numerical analysis it is common to approximate a solution to a PDE $$u'(t) = (A+B) u(t), \quad u(0)=u_0$$ which is just given by $e^{t(A+B)}u_0$ by the splitting $e^{tB/2} e^{tA} e^{tB/2}u_0.$ Here,...
Sascha's user avatar
  • 536
5 votes
0 answers
208 views

Perturbation of Neumann Laplacian

Consider the $N \times N$ matrix $$A_{\alpha}=\begin{pmatrix} \lambda_1 & -1 & -\alpha & 0 & \cdots & 0\\ -1 & \lambda_2 & -1 & -\alpha & \cdots & 0\\ -\alpha &...
Guido Li's user avatar
4 votes
2 answers
871 views

Decay of eigenfunctions for Laplacian

Consider the discrete second derivative with Dirichlet boundary conditions on $\mathbb C^n$. Its eigendecomposition is fully known: see wikipedia It seems like the largest eigenvalue $\lambda_1$ is ...
Yannis Pimalis's user avatar
3 votes
0 answers
163 views

Perturbation theory compact operator

Let $K$ be a compact self-adjoint operator on a Hilbert space $H$ such that for some normalized $x \in H$ and $\lambda \in \mathbb C:$ $\Vert Kx-\lambda x \Vert \le \varepsilon.$ It is well-known ...
user avatar
3 votes
1 answer
791 views

Real part of eigenvalues and Laplacian

I am working on imaging and I am a bit puzzled by the behaviour of this matrix: $$A:=\left( \begin{array}{cccccc} 1 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & -1 &...
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2 votes
0 answers
331 views

What is the spectrum of this differential operator?

My self-adjount differential operator $L$ is defined by $$L f(x) \equiv u(x) \frac{\partial^2}{\partial x^2} \left( u(x) f(x) \right)$$ where $u(x)$ is a known but arbitrary smooth function that ...
Peter A's user avatar
  • 151
1 vote
0 answers
70 views

Lower-bounding the eigenvalues of a certain positive-semidefinite kernel matrix, as a function of the norm of the input matrix

Let $\phi:[-1,1] \to \mathbb R$ be a function such that $\phi$ is $\mathcal C^\infty$ on $(-1,1)$. $\phi$ is continuous at $\pm 1$. For concreteness, and if it helps, In my specific problem I have $\...
dohmatob's user avatar
  • 6,853
0 votes
0 answers
74 views

Computing the eigenvalues of $A+E$ where $A$ is an upper triangular matrix whose diagonal entries are all zero and $E$ is a rank one matrix

Let us consider the backward-shift matrix $B=(b_{ij})\in M_n(\mathbb{R})$ whose entries are given by $b_{k,k+1}=1$ and the other entries are all 0. We also consider $X=(x_{ij})\in M_n(\mathbb{R})$ ...
ABB's user avatar
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0 votes
0 answers
104 views

Necessary and sufficient conditions for $\mbox{trace}(A^{-1/2}e^{-tB} (AB+BA) e^{-tB}A^{-1/2}) \ge 0$ for all $t$

Let $A$ and $B$ be positive-definite matrices of the same size. For any $t \ge 0$, define $$ u(t) := \mbox{trace}(A^{-1/2}e^{-tB} (AB+BA) e^{-tB}A^{-1/2}). $$ Question. What are necessary and ...
dohmatob's user avatar
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