solving $A=BCD$ for $C$, where $A,B,C,D$ are all matrices

Question

If $A,B,C,D$ are all matrices and $A=BCD$ (with dimensions such that all matrix multiplications are defined), how does one solve for $C$?

In the particular context I'm working in, $B$ and $D$ are both orthogonal, and $C$ is diagonal. I'm not sure if that's necessary to solve for $C$.

Answer 1

You would have to multiply by $B^{-1}$ on the left on each side of the equation which would cancel $B$ on the right, and then by $D^{-1}$ on the right on each side, which would cancel $D$, like this: $$\begin{align}A&=BCD\\B^{-1}A&=B^{-1}BCD\\B^{-1}A&=CD\\B^{-1}AD^{-1}&=CDD^{-1}\\B^{-1}AD^{-1}&=C\end{align}$$

Answer 2

Hint:

• $B^TB=I$

• $DD^T = I$

• You might like to premultiply and postmultiply the equation by some matrices to isolate $C$.

Answer 3

Since $B, D$ are orthogonal,

$B^t B = D D^t = I$

multiply your equation with $B^t$ from the left and by $D^t$ from the right to get $$C = B^t A D^t$$