(1) |
This identity is not very intuitive and so we will provide the proof here. This proof depends upon one assumption being true, that C = C^{T}.
Let us define,
(2) |
and,
(3) |
We start by expanding X,
X | (A - B C^{-1}) C (A - B C^{-1})^{T} - B C^{-1} B^{T} | ||
= | (A - B C^{-1}) C (A^{T} - C^{-T}B^{T}) - B C^{-1} B^{T} | (4) |
X | = | (A - B C^{-1}) C (A^{T} - C^{-1}B^{T}) - B C^{-1} B^{T} | |
= | (A C - B ) (A^{T} - C^{-1}B^{T}) - B C^{-1} B^{T} | ||
= | ACA^{T} - BA^{T} - AB^{T} + BC^{-1}B^{T} - BC^{-1}B^{T} | ||
= | (5) |