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I have a matrix $$ A= \begin{pmatrix} 0 & a & d & c\\ \bar a & 0 & b & d \\ \bar d & \bar b & 0 & a \\ \bar c & \bar d & \bar a & 0 \end{pmatrix} $$ As you can see, the matrix is always self-adjoint …

Normal compact linear operators on Hilbert spaces have infinitely many (counting multiplicities) eigenvalues by the spectral theorem. For non-normal operators this no longer has to be true. There exis …

Playing around with Matlab I noticed something very peculiar: Take the symmetric matrix $A \in \mathbb R^{n \times n}$ defined by $$A_{ij}= i \delta_{ij} - \frac{\varepsilon}{\sqrt{i}\sqrt{j}}\,.$$ …

Let $\mathbb{C} \ni z \mapsto M(z)$ be a square matrix depending holomorphically on a parameter $z$ with the property that $\operatorname{dim}\ker(M(z)))=1$ for $z $ away from a discrete set $D \subse …

We consider block matrices $$\mathcal A = \begin{pmatrix} 0 & A\\A^* & 0 \end{pmatrix}$$ and $$\mathcal B = \begin{pmatrix} 0 & B\\C & 0 \end{pmatrix}.$$ Then we define the new matrix $$T(t) = \begin{ …

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 …

I make the following observation: Let $\Delta^{(n)}$ be the discrete Laplacian on $\mathbb{C}^n$ (ie the $n\times n $ matrix with diagonal $-2$ and upper/lower diagonal $1$.) This one has eigenvalues …

In a previous question on mathoverflow, I asked about the following: Let $\Delta$ be the Laplacian on some compact interval $I$ of the real line with let's say Dirichlet boundary conditions. The funct …

A Cauchy-Euler operator is an operator that leaves homogeneous polynomial of a certain degree invariant, named after the Cauchy-Euler differential equations We consider the operator $$(Lf)(x) = \langl …

I edited the question to a general mathematical question, since I found the answer in Carlo Beenakker's reference and think that my initial question was mathematically misleading. What was behind al …

Consider the following block matrix $$A = \left(\begin{matrix} B & T\\ T & 0 \end{matrix} \right)$$ where all submatrices are square and matrix $B = \mbox{diag}\left(b_1 ,0,0,\dots,0,b_n \right)$ …

In a previous question on mathoverflow, I was wondering about the following: Let $\Delta$ be the Laplacian on some compact interval $I$ of the real line with let's say Dirichlet boundary conditions. A …

Consider the matrix-valued operator $$A = \begin{pmatrix} x & -\partial_x \\ \partial_x & -x \end{pmatrix}.$$ I am wondering if one can explicitly compute the eigenfunctions of that object on the spac …

This is a follow up question on my previous question here that was on solved in the deterministic setting by Denis Serre, when the perturbation can be separated. Therefore, I decided to split the det …

Consider the linear constant coefficient differential operator $P$ on the Hilbert space $L^2([0,1]^2;\mathbb C^2)$ $$P= \begin{pmatrix} D_{z}+c & a \\ b & D_{z}+c \end{pmatrix}$$ where $D_z=-i \parti …