Hi all! I have been banging my head for this one. Can anyone see an error or a way out of the wilderness?
x is an observation vector. m1, m2 are both mean vectors. S1 is inverse of the covariance matrix. (') means transpose.
Question: Show (I always start sweating when questions start that way)
that
(-1/2)(x-m1)'S1(x-m1) + (1/2)(x-m2)'S1(x-m2)
=
(m1-m2)'S1x - (1/2)(m1-m2)'S1(m1-m2).
I have expanded the left hand side and gotten
(-1/2) [x'S1x - m1'S1x - x'S1m1 + m1'S1m1 - x'S1x + m2'S1x + x'S1m2 -m2S1m2 ]
I canceled x'S1x - x'S1x so get
(-1/2) [- m1'S1x - x'S1m1 + m1'S1m1 + m2'S1x + x'S1m2 -m2S1m2 ]
Then saw -m1'S1x +m2'S1x could be re-written (-m1+m2)S1x so I have now:
(-1/2) [(-m1+m2)S1x- x'S1m1 + m1'S1m1 + x'S1m2 -m2S1m2 ]
OR
(-1/2) [(-1)*(m1-m2)S1x- x'S1m1 + m1'S1m1 + x'S1m2 -m2S1m2 ]
But now Im stuck and cant see where to proceed.
Any help?
Thanks!
Brian