SORRY FOR POORLY FRAMED QUESTION. JUST SIGNED UP HERE. THANK YOU.
I am trying to derive the discriminant function for a problem given in Pattern Recognition. I am stuck at following step:
$$x^{T}\Sigma_{i}^{-1}\mu_{i}=\mu_{i}^{T}\Sigma_{i}^{-1}x$$
I found this equation on this wikipedia page on Linear dicriminant analysis. It says that the equation is valid as the $Σ_i$ matrix is Hermitian. How do I prove this?
The author has used "Hermitian" when "symmetric" is more appropriate. The equation doesn't use the Hermitian transpose, so it wouldn't be correct as written for complex vectors/matrices. Normally, the covariance matrix $\Sigma$ would be real, so I'll proceed by assuming that everything in the equation is real.
The quantity on the left hand side of the equation is a scalar, so it is equal to its transpose.
$x^{T}\Sigma_{i}^{-1}\mu_{i} = (x^{T}\Sigma_{i}^{-1}\mu_{i})^{T}$
The transpose of the product is the product of the transposes in reverse order.
$ (x^{T}\Sigma_{i}^{-1}\mu_{i})^{T} = \mu_{i}^{T}\Sigma_{i}^{-T}x$
Since $\Sigma_{i}^{-1}$ is symmetric, $\Sigma_{i}^{-T}=\Sigma_{i}^{-1}$, and
$x^{T}\Sigma_{i}^{-1}\mu_{i} = \mu_{i}^{T}\Sigma_{i}^{-1}x$