$A_{n \times m} D_{m \times m} A^T_{m \times n} + \alpha I_{n \times n}$

Question

Assume that we have a matrix product of form $B=A_{n \times m} D_{m \times m} A^T_{m \times n} + \alpha I_{n \times n}$. $D$ is a positive diagonal matrix, $I$ is identity matrix, $\alpha>0$ and $m < n$.

Is there any relation between $|B|$, $|AA^T|$, $|D|$ and $|I|$ or their logarithms, in the form of an equality or an inequality?

Answer 1

\begin{align} \det(A.D.A^\top + \alpha I) &= \sum_{k=0}^n \alpha^{n-k} \text{Trace}(\Lambda^k(A.D.A^\top)) \\& = \sum_{k=0}^n \alpha^{n-k} \text{Trace}\big(\Lambda^k(A).\Lambda^k(D).\Lambda^k(A^\top)\big) \end{align}