Let $A$ be real symmetric matrix. It is a well-known observation that we can bound any eigenvalue $\lambda$ of $A$ by using the fact that $$\lambda^{2 k} \leq \textrm{Tr} A^{2 k}$$ for any $k\geq 1$. If we know that $\lambda$ has multiplicity at least $M$, then, evidently, we can say something stronger: $$M\cdot \lambda^{2 k} \leq \textrm{Tr} A^{2 k}.$$
Question: how old is each of these two observations? Are they perhaps as old as matrices (Sylvester, Cayley)? Vaguer question: for how long have these observations been the basis of a common strategy for bounding eigenvalues, and in which fields?
PS. I originally included a "graph-theory" tag, which is too general and also too specific. My question is not limited to graph theory, where these observations are bread-and-butter; the case of fields where this strategy is less common (but still fruitful) may be at least as interesting.
