I have a matrix equation $A \cdot B \cdot C = D$ with $A$ being $1\times n$ row vector, $B$ a $n\times n$ matrix and $C$ a $n\times1$ column vector. $D$ is a number.
I want to calculate $n\times n$ matrix $\Pi$ = $\dfrac{A \otimes C}{D}$ with $\otimes$ being the outer product. I understand it involves eigenvectors of matrix $B$.
Can someone guide me how to calculate either $\Pi$ or solve the matrix equation for $B$?
Doing it by inversion, i.e. multiplying both sides by $B^{-1}$ does not work due to dimensions of $A$ and $C$.
Here's the background: I am going through Gerald. D. Mahan's book (many-particle physics). In Sec. 5.5 (in 2nd edition),the dielectric function is calculated by Heisenberg equation of motion.
However, in a multi-orbital (or pseudo-spin, for example) case, the dielectric function becomes a tensor and the defining equation (3rd equation in page 433) takes the form $A_{1 \times n} \cdot B_{n \times n} \cdot C_{n \times 1} = D$. Consequently, the density response function becomes a tensor which has the form $\Pi$ = $\dfrac{A \otimes C}{D}$.