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Suppose $S_0, S$ are known real symmetric matrices and we have the function $f(x) := \lambda_{0}(S_0+x S)$ where $\lambda_0$ is the smallest eigenvalue.

By symmetry, the eigenvalues are real and I believe the smallest eigenvalue is a smooth function of $x$. How can I find the derivative $\frac{df(x)}{dx}$

Lewwwer
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  • Glancing at this answer, it seems the answer might be $v_0^\top S v_0$ where $v_0$ is the unit eigenvector corresponding to the smallest eigenvalue of $S_0$, as long as you satisfy some additional assumptions (uniqueness of the smallest eigenvalue of $S_0$?). I may be missing some details though. – angryavian Mar 27 '22 at 16:04

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Yes, since the matrices $S_0$ and $S$ are symmetric, then the matrix $S_0+xS$ is diagonalizable, which means that all the eigenvalues are semisimple and, as a result, that they are smooth functions of the parameters.

If the minimum eigenvalue $\lambda_0(S_0)$ is simple, then the derivative is simply given by $v_0^TSv_0$ where $v_0$ is the eigenvector of $S_0$ associated with the eigenvalue $\lambda_0(S_0)$.

When the minimum eigenvalue $\lambda_0(S_0)$ is not simple, then we have that the derivative of all those eigenvalues to be the eigenvalues of $V_0^TSV_0$ where $V_0$ consists all the eigenvectors of $S_0$ associated with the minimum eigenvalue of $\lambda_0(S_0)$.

However, note that those results are only valid in a neighborhood of $x=0$. More generally, if you want to behavior of the eigenvalues around the point $x_0$, then you will need to substitute $S_0$ by $S_0+x_0S$ in the above expressions.

Reference: Seyranian, Mailybaev, "Multiparameter Stability Theory with Mechanical Applications"

KBS
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  • Thank you. So basically it is $\operatorname{inf}{v_0 \in V_0, ||v_0||=1} \ \ v{0}^{T}S v_{0}$ where $V_0$ is the eigenspace for the smallest eigenvalue of $S_0$ – Lewwwer Mar 29 '22 at 11:56
  • That will give you the smallest variations (not in absolute value) for the smallest eigenvalues. Note that this is only local, another eigenvalue can still take smaller values for some $x>0$. – KBS Mar 29 '22 at 12:41