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)),

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