Is $\det(I+aAVV^*A^*)$ increasing function in $a$. Is $$\det(I+aAVV^*A^*)$$ increasing function in $a$?
Here $A$, $V$ are complex matrices, $a$ is positive real value, $I$ is identity matrix and $^*$ defines conjugate transpose (Hermitian).
Thank you.
PS. NOT home work, related to my research work in information theory.

Let $X=AVV^*A^*$. Then $X$ is positive semi-definite. So its eigenvalues $x_1,\ldots,x_n$ are all non-negative. 
Then we have
$$
\det(I+aX)=\prod_{j=1}^n(1+ax_j).
$$
If $a<b$, then $1+ax_j<1+bx_j$, since $x_j\geq0$. The condition $a>0$ guarantees that all factors $1+ax_j$, $1+bx_j$ are positive. So
$$
\det(I+aX)=\prod_{j=1}^n(1+ax_j)\leq\prod_{j=1}^n(1+bx_j)=\det(I+bX).
$$
If we allow $a$ to be negative, the situation is more complicated, as different factors might have different signs. 
