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In my reaseach, recently, I came across of this problem where I have to compute, analytically, the derivative of the dominant eigenvalue of the following matrix.

Let $D$ be a diagonal real $n \times n $ matrix $\text{diag} \{d_{1}, \dots , d_{n}\}$. Let A be an essentially nonnegative matrix (that is $a_{ij}\geq 0$, for all $i\neq j$). It is known that $r(A+D)$ is a convex function of $D$ (see ref), where $r(A+D)$ is the dominant eigenvalue of an $A+D$.

I need to find an expression for $\frac{\partial r(A+D)}{\partial D_{jj}}|_{D^{*}} $, where $D^{*}$ is a diagonal matrix. It would be great if someone can direct me on how to compute this derivative.

ref: @article{cohen1981convexity, title={Convexity of the dominant eigenvalue of an essentially nonnegative matrix}, author={Cohen, Joel E}, journal={Proceedings of the American Mathematical Society}, volume={81}, number={4}, pages={657--658}, year={1981} }

Thank you.

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  • $\begingroup$ Even in the $2x2$ case, this seems to be quite a complicated expression. $\endgroup$
    – Steven Landsburg
    Commented Nov 9, 2013 at 0:06
  • $\begingroup$ There was a typo in my question previously. I have edited it. I was hoping that i could get an expression for the derivative in terms of the eigenvectors of $(A+D)$ corresponding to the eigenvalue $r(A+D)$. $\endgroup$
    – gandufen maths
    Commented Nov 9, 2013 at 1:15
  • $\begingroup$ Link to the referenced article: lab.rockefeller.edu/cohenje/PDFs/… $\endgroup$
    – Todd Trimble
    Commented Nov 9, 2013 at 2:01

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Check out this paper:

Derivatives of the Perron root at an essentially nonnegative matrix and the group inverse of an $M$-matrix

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