How to solve this equation in $\lambda$?

Let

$$\mathbf{u} := \left( \mathbf{X}^H \mathbf{X} + \mathbf{I}_m + \mathbf{\lambda}\mathbf{D} \right)^{-1} \mathbf{X}^H \mathbf{y}$$

where

• $$\mathbf{X}$$ is $$n \times m$$ semi-orthogonal matrix ($$\mathbf{X} \mathbf{X}^H = \mathbf{I}_n$$)

• $$\mathbf{D}$$ is $$m \times m$$ diagonal matrix

• $$\mathbf{y}$$ is $$n \times 1$$

• $$\mathbf{I}_m$$ is $$m \times m$$ identity matrix

and $$m \gg n$$. Let

$$\rho(\lambda) := \mathbf{u}^H \mathbf{u}$$

Find $$\lambda > 0$$ such that $$\rho ' (\lambda) = 0$$

Appreciate any solutions or any numerical schemes for finding $$\lambda$$.

EDIT

Working on the lines of the comments, I get expression for the derivative.

Let $$G = X^HX+I+\lambda D$$, assuming all entries of $$D$$ are real (as I am interested in it(I forgot to mention before)),

$$\frac{\partial \rho}{\partial \lambda} = -y^HXG^{-1}\Big(DG^{-1} + G^{-1}D\Big)G^{-1}X^Hy$$

• In general $\frac{\partial A^{-1}}{\partial \lambda}=A^{-1}\frac{\partial A}{\partial\lambda}A^{-1}$. With this it should be possible to get an expression for $\frac{\partial \rho}{\partial \lambda}$. – user100927 Apr 26 at 9:32
• @user100927 : Thanks. I have done that. I don't know how to solve from there, the matrix equation for the scalar $\lambda$. – Rajesh Dachiraju Apr 26 at 9:48
• @RodrigodeAzevedo : have added that. – Rajesh Dachiraju Apr 29 at 8:59
• How large are $m$ and $n$? Why not use symbolic computation? Brute, but it may work. – Rodrigo de Azevedo Apr 30 at 8:48
• @RodrigodeAzevedo : I am in search of an algorithm to compute $\lambda$, given the matrices. The sizes $n$ and $m$ could be arbitrary, only condition being $m > n$. Also generally $m >> n$, if that helps to reduce algorithm complexity. – Rajesh Dachiraju Apr 30 at 9:26