I am currently studying a system characterized by a particularly structured matrix, as shown below, $$ \dot{ \boldsymbol{x}} = \begin{pmatrix} 0 & -2c & \\ c & 0& 3c & \\ &-2c & 0& -4c & \\ & & 3c & 0 & 5c \\ & & & \ddots & \ddots &\ddots \end{pmatrix} \boldsymbol{x}.$$ Let's denote this matrix as $cF$. I am particularly interested in the cases where $F$ is $n \times n$ and when it extends to an infinite-dimension. My objectives are to understand any special naming conventions for such matrices and their eigenvalue properties, especially as the dimension varies.
- Naming and Known Properties: Does this matrix, or ones with a similar structure, have a special name or known properties, such as eigenvalues for the finite-dimensional case? I attempted to factorize $F$ as shown below but didn't manage to derive any conclusion on eigenvalue properties from this factorization: $$ F=\begin{pmatrix} 0 & -2 & \\ 1 & 0& 3 & \\ &-2 & 0& -4 & \\ & & 3 & 0 & 5 \\ & & & \ddots & \ddots &\ddots \end{pmatrix}=\begin{pmatrix} 0 & 1 & \\ 1 & 0& 1 & \\ 0 & 1 &\ddots &\ddots\\ & & \ddots & \end{pmatrix}\begin{pmatrix} 1 \\ & -2 \\ & & 3 \\ & & & -4 \\ & & & & \ddots \end{pmatrix} =BD. $$
And I know that $DB=F^\top$. But I could not conclude anything from this expression. I also observed (numerically) that the finite matrix seems to have purely imaginary eigenvalues.
- Analytical Tools for Infinite-Dimensional Analysis: Considering the difference in the spectrum between finite and infinite-dimensional systems, what analytical tools or approaches could be recommended to study or bound the eigenvalues of the infinite-dimensional system? Any insights, suggestions, or references to relevant literature would be extremely helpful and appreciated!