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 $i\neq j$). I 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 \rho(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.

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.

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 $i\neq j$). I 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 \rho(DA)}{\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.

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 $i\neq j$). I 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 \rho(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.