Is there an explicit relationship between the eigenvalues of a matrix and its derivative? If we consider a matrix $A$ dependent of a variable $x$, the eigenvalues and eigenvectors satisfying the equation
$$
A \vec{v}=\lambda \vec{v}
$$

will also depend on $x$. If we consider the matrix $B$ such that 
$$B_{ij}=\frac{ \mathrm{d}}{ \mathrm{d} x} A_{ij}$$ 
Then, could we express the eigenvalues of $B$ in terms of the eigenvalues of $A$?
I found the question very interesting and was not able to find a satisfying answer myself.

For example in the case for $2\times2$ matrices of the form
$$
A=\left (
\begin{matrix}
 a(x) &amp; b(x) \\ 
 0 &amp; c(x) 
\end{matrix}
\right ),\implies B=\left (
\begin{matrix}
 a'(x) &amp; b'(x) \\ 
 0 &amp; c'(x) 
\end{matrix}
\right )
$$
I noticed that $\lambda_B(x)= \lambda_A'(x)$. But I cannot generalise it to general $2\times 2$ matrices. Not even thinking about $n\times n$ matrices...

Thank you for your help and any idea!
It is not true in general that the eigenvalues of $B(x)$ are the derivatives of those of $A(x)$. And this even for some square matrices of dimension $2 \times 2$.

Consider the matrix

$$A(x) =
\begin{pmatrix}
1&amp; -x^2\\
-x &amp;1
\end{pmatrix}$$
It’s characteristic polynomial is $\chi_{A(x)}(t)=t^2-2t+1-x^3$, which has for roots $1\pm x ^{3/2}$ for $x&gt;0$. Those are the eigenvalues of $A(x)$.

The derivative of $A(x)$ is 
$$B(x) =
\begin{pmatrix}
0&amp; -2x\\
-1 &amp;0 
\end{pmatrix}$$ and it’s characteristic polynomial is $\chi_{B(x)}(t)=t^2-2x$, whose roots are $\pm \sqrt{2} x^{1/2}$for $x&gt;0$.

We get a counterexample as the derivative of $1+x^{3/2}$ is not $\sqrt{2}x^{1/2}$.

However in the special case of upper triangular matrices (that you consider in your original question) the eigenvalues of the matrix derivative are indeed the derivatives of the eigenvalues.
If $A(x)v= \lambda(x)v$, with v independent of x, then, differentiating on both sides of the equation by x, $A'(x)v= \lambda'(x)v$.  That is, the eigenvalues of A' are the derivatives of the eigenvalues of A, with same associated eigenvectors.