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Consider the matrix valued function $A: \mathbb R \rightarrow \mathbb R^{n \times n}$. For a particular $x \in \mathbb R$, $A(x)$ has Eigenvalues $\lambda_1, \dots, \lambda_n$. Is it possible to compute the sensitivities of the eigenvalues w.r.t. $x$? I.e.

$$ \frac{\partial \lambda_i}{\partial x}. $$

I am aware of the standard Eigenvalue perturbation theory as in https://en.m.wikipedia.org/wiki/Eigenvalue_perturbation, but there the matrix is symmetric and with a perturbation that is necessarily a scaled version of the matrix (e.g. $A+\delta A$), which is not the case in this setup.

Thanks in advance!

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  • $\begingroup$ The perturbation is not necessarily a scaled version of the matrix... The notation $\delta A$ does not stand for a number $\delta$ multiplied by matrix $A$, it is the perturbation matrix. The perturbed matrix is $\tilde A = A + \delta A$ and thus $\delta A = \tilde A - A$. $\endgroup$ Commented Oct 22, 2024 at 7:30

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They are essentially the same: Consider a simple version that $A=A_0+xB$ and $A_0=\text{diag}(\lambda_1,\dots,\lambda_n)$, with all eigenvalue multiplicity 1. Then $$\left.\frac{\partial \lambda_i}{\partial x}\right\vert_{x=0}=B_{ii}$$.

Proof: Look at the characteristic polynomial, only diagonal would contribute to first order term of $x$

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