# The relation of eigenvalues in linear algebra and Sturm-Liouville problems

Is there a relation between eigenvalues of a matrix and the eigenvalues of a differential operator in a Sturm-Liouville problem? The two problems are denoted by $$AV=\lambda V$$ and $$Lf=\lambda f$$ respectively. where $$A$$ is a matrix, $$L$$ is a differential second order operator, $$V$$ is eigenvector, $$f$$ is eigenfunction and $$\lambda$$ is eigenvalue. One can see what has transformed into what. If the two problems are related, is there a similar approach to matrix problem which can be used in Sturm-Liouville problem in order to derive the eigen-values?

• Believe it or not, the differential equation eigenfunction came first, before the matrix case. The first derived from looking at the problem of a vibrating string in the 1700's. The matrix case came much later. Commented Sep 28, 2022 at 15:24
• @DisintegratingByParts: that is something that always fascinated me. I learned about it in "History of functional analysis" of Dieudonné. You are probably already very familiar with this reference. Still, I'd like to mention the introduction of that book, which says "the evolution of linear algebra has been slow and painful, [...] in a succession of stages which [...] is exactly the reverse of the logical sequence of notions [...]". books.google.pt/… Commented Sep 29, 2022 at 10:14