As noted in the comments, the following notations will be used:
- $\lambda_i$ is the $i^{th}$ eigenvalue (presumably listed by descending magnitude)
- $|v_i\rangle$ is the $i^{th}$ column eigenvector
- $\langle v_i|$ is the $i^{th}$ row eigenvector
Suppose I have a large matrix $A(x,t)$ depending on two free parameters (so each input of $A$ is a continuous function of $x$ and $t$). Let $$ B(x,t)=\frac \partial {\partial t}A(x,t)\\ C(x,t)=\frac \partial {\partial x}B(x,t) $$ If I know all the eigenvalues and eigenvectors of $A$, and I know all the eigenvalues and eigenvectors of $C$, then can I use that information to find the complete eigenvalues and eigenvectors of $B$?
I know that if such a relationship exists that it is non-trivial, since, for example, if $$ A=\sum \lambda_i |v_i\rangle\langle v_i|, $$ then $B$ is given by $$ B=\sum (\frac \partial {\partial t}\lambda_i) |v_i\rangle\langle v_i|+ \lambda_i |\frac \partial {\partial t}v_i\rangle\langle v_i|+ \lambda_i |v_i\rangle\langle \frac \partial {\partial t}v_i|, $$ which shows that $B$ is not diagonalized the same as $A$, but surely a relationship must exist between them.