$\mathbf{I}$ is the $K\times K$ identity matrix.
$\mathbf{h}_i\in\mathbb{C}^{M\times1}\quad\forall1\leq i\leq K$ are column vectors.
Consider the solution of the convex optimisation problem over $\mathbf{x}=[x_1, x_2, \cdots, x_K]$ $$\mathbf{x}^*=\mbox{argmax}\log|\mathbf{I}+\sum_{i=1}^Kx_i\mathbf{h}_i\mathbf{h}_i^{\text{H}}|$$
under the constraints $x_i\geq0$ and $\sum_{i=1}^Kx_i=P$. Show that $\mathbf{x}^*$ is an element-wise non-decreasing function of $P$.
P. S. Basically the objective function is the multiple access channel data rate under a power constraint. I am trying to prove that if the available total power is increased, the optimally available power increases for each user $x_i$. This is intuitively obvious, but can't prove it.