Question

Assuming that all matrices are n x n and invertible, solve for D .

C B (A^T) D B (C^T) A = C B^T

Answer 1

First multiply by C inverse on the left\[C ^{-1}CBA ^{T}DBC ^{T}A = C^{-1}CB ^{?}\]
Yielding:
\[BA ^{T}DBC ^{T}A = B ^{T}\]

Then Multiply by B inverse and A transpose inverse on the left yielding

\[DBC ^{T}A=B ^{-1}A^{-T}B^{T}\]

Now, multiply by B inverse, C transpose inverse and A inverse on the right

\[D=B ^{-1}A^{-T}B^{T}B^{-1}C^{-T}A^{-1}\]