42
votes

Accepted

### Intuitively, what does a graph Laplacian represent?

How to understand the Graph Laplacian (3-steps recipe for the impatients)
read the answer here by Muni Pydi. This is essentially a concentrate of a comprehensive article, which is very nice and well-...

38
votes

Accepted

### Spectral symmetry of a certain structured matrix

For real $a,b,c$ and imaginary $d$ the matrix $A$ has chiral symmetry, meaning it anticommutes with a matrix $X$ that squares to the identity:
$$X=\left(
\begin{array}{cccc}
0 & 0 & 0 & -...

37
votes

### Can you hear the shape of a drum by choosing where to drum it?

Sometimes you can
Since there seem to be no hard results along these lines on the literature, I decided to have a look at the two shapes in the question and see if there are some relatively ...

32
votes

Accepted

### Why should I look at the resolvent formalism and think it is a useful tool for spectral theory?

Preliminary remark. As mentioned in the comments, I find the notion "resolvent formalism", as well as the description in the Wikipedia article, rather misleading - resolvents are not ...

25
votes

Accepted

### Can one hear the (topological) shape of a drum?

There are examples due to Ikeda of isospectral Lens spaces which are not homotopy equivalent.
Likeliest the simplest examples are the compact connected 3-dimensional flat manifolds which are a ...

24
votes

Accepted

### How are eigenvalues and eigenvectors affected by adding the all-ones matrix?

This is a special case of a rank one perturbation or a rank one update, and there is plenty of work on such. See the nice 2010 lecture notes by Andre Ran.

23
votes

Accepted

### Eigenfunctions of the laplacian on $\mathbb{CP}^n$

The $k$-th eigenfunctions are actually easy to describe: In $\mathbb{C}^{n+1}$ with unitary complex coordinates $z_0,z_1,\ldots,z_n$, write $Z = |z_0|^2+\cdots+|z_n|^2$.
Now, for a given $k\ge0$, ...

22
votes

### Why is there no symplectic version of spectral geometry?

From a certain point of view the premise of the question is wrong. The study of sympletic manifolds with no additional structure is akin to differential topology rather than differential geometry. ...

20
votes

Accepted

### Why is there no symplectic version of spectral geometry?

The characteristic variety (i.e. vanishing locus of the symbol) of a symplectomorphism invariant scalar differential equation is a real projective hypersurface invariant under the group of ...

20
votes

Accepted

### Eigenvalue pattern

The explanation is pretty simple with a suitable change of basis.
Letting
$$B =
\begin{pmatrix}
1 & 0 & 1 & 0 \\
i & 0 & -i & 0 \\
0 & 1 & 0 & 1 \\
0 & i &...

19
votes

Accepted

### Non real eigenvalues for elliptic equations

Here is a construction. It elaborates from perturbation analysis of eigenvalues. However it starts from the situation of a non-simple eigenvalue.
So, let me start with the standard self-adjoint $L_0=-\...

18
votes

### Reasons for difficulty of Graph Isomorphism and why Johnson graphs are important?

Johnson graphs do not cause difficulty to existing programs. Actually they are rather easy; nauty can handle them up to tens of millions of vertices, and so can other programs such as Traces and Bliss....

18
votes

### Rigorous justification for this formal solution to $f(x+1)+f(x)=g(x)$

A direct and precise way to arrive at your answer is to appeal to the theory of Fourier transforms of distributions. If I define the Fourier transform as
$$F(k)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\...

17
votes

Accepted

### Differentiability of eigenvalues of positive-definite symmetric matrices

In the open subset of $M_n(\mathbb{R})$ where the $\lambda_i$ are distinct, they are $C^{\infty}$ functions: this follows from the implicit function theorem.
On the other hand, when some eigenvalue ...

17
votes

### Intuitively, what does a graph Laplacian represent?

This is just a long comment, adding to the excellent answers above.
There is a great article from László Lovász "Discrete and Continuous:
Two sides of the same?", written around 2000 (https:/...

16
votes

### Differentiability of eigenvalues of positive-definite symmetric matrices

The keyword is the Cartan decomposition in the theory of symmetric spaces.
In short, when an eigenvalue is simple (its multiplicity is $1$) it is locally an analytic function. But when the ...

16
votes

### Spectral symmetry of a certain structured matrix

An equivalent trick : Let $J:= \operatorname{diag}(1,i,-1,-i)$. Then $J^*AJ=iB$ where $B$ is real and skew-symmetric. Hence the spectrum of $iB$ (thus that of $A$) comes by pairs $\pm\lambda$.

15
votes

Accepted

### An orbit of symmetric polynomials

Suppose $f(x,y,z)$ is symmetric (in the following, symmetric tout court always means "symmetric w.r.to the three variables $(x,y,z)$") . Then $\mathcal{L}f(x,y,z):=(x+y)(x+z)f(x-1,y,z)-x^2f(x,y,z)$ ...

15
votes

Accepted

### Imaginary eigenvalues

Define the unitary and Hermitian matrices
$$U=\left(
\begin{array}{cccc}
0 & 0 & -i & 0 \\
0 & 0 & 0 & -i \\
i & 0 & 0 & 0 \\
0 & i & 0 & 0 \\
\end{...

14
votes

### Spectrum of $L^\infty(X,\mu)$

I'm recording here some references on the object $\tilde X = \mathrm{Spec}(L^\infty(X,\Sigma,\mu))$ (many of which were communicated to me recently by Balint Farkas), assuming for sake of simplicity ...

14
votes

Accepted

### Eigenvalues of Laplace-Beltrami on half sphere

Using symmetry, you can extend any Dirichlet eigenfunction on the upper half-sphere to the entire sphere $\mathbb{S}^{N-1}$. Therefore, the spectrum of the upper hemisphere is a subset of the spectrum ...

14
votes

### Algebraic graph invariant $\mu(G)$ which links Four-Color-Theorem with Schrödinger operators: further topological characterizations of graphs?

Perhaps I can contribute to the history part the question, since I was quite close to the Institut Fourier at that time and was very interested in their work (I am a physicist). Grenoble now has ...

14
votes

### Intuitively, what does a graph Laplacian represent?

This is not really about the connection with graph theory, a topic I am rather ignorant of, but rather the connection to continuum notions, all of which I learned from this paper.
Consider a ...

14
votes

Accepted

### Spectrum of matrix involving quantum harmonic oscillator

The Hamiltonian
$$H=\begin{pmatrix}
\alpha^2+a^\ast a&\alpha a+\beta a^\ast\\
\alpha a^\ast+\beta a&\beta^2+a^\ast a
\end{pmatrix}
$$
is known in the physics literature as the anisotropic Rabi ...

13
votes

### Can you hear the shape of a drum by choosing where to drum it?

As mentioned in the comments, knowing both eigenvalues and eigenfunctions gives you enough information to find the shape of the domain, so to make this problem more challenging one might ask what ...

13
votes

### Eigenvalues of the Laplace-Beltrami operator on a compact Riemannnian manifold

A modern "simple philosophical" explanation is that this problem can be restated as an eigenvalue problem for a compact operator in an appropriate Hilbert space, whose eigenvalues are reciprocal to ...

13
votes

### Differentiability of eigenvalues of positive-definite symmetric matrices

As mentionned by other answers, simple eigenvalues are $C^\infty$, while non-simple ones are not. Let me add however two important properties which you can find in Kato's book Perturbation theory of ...

12
votes

### How are eigenvalues and eigenvectors affected by adding the all-ones matrix?

A cute fact that is trivial to prove is this: define the characteristic polynomial of a matrix $M$ by $\phi_M(x) = |xI-M|$. Then for any $A$ and any $s$, $$\phi_{A+sJ}(x) = (1-s)\phi_A(x)+s\phi_{A+J}(...

12
votes

### Intuitively, what does a graph Laplacian represent?

I wrote a blog post a while ago* on different ways of interpreting the graph laplacian from the perspectives of functional analysis, probability, statistics, differential equations, and topology, and ...

12
votes

### Why is the length spectrum called a spectrum?

I do not know of such an operator. But there is the following lovely theorem of Huber:
Theorem: Two compact hyperbolic surfaces have the same spectrum of the Laplacian if and only if they have the ...

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