Maximize $-\frac{1}{2}(x-\mu_k)^T\Sigma^{-1}(x-\mu_k)$ over $k$

Question

Can someone point out the development from Eq(1) to Eq(2) of the following maximization over $k$ ?

Eq(1): $-\frac{1}{2}(x-\mu_k)^T\Sigma^{-1}(x-\mu_k)$

Eq(2): $C -\frac{1}{2}\mu_k^T\Sigma^{-1}\mu_k + x^T\Sigma^{-1}\mu_k$ where $C$ denotes a constant

Answer 1

$$(x-\mu_k)^T\Sigma^{-1}(x-\mu_k) = x^T\Sigma^{-1}x - x^T\Sigma^{-1}\mu_k - \mu_k^T\Sigma^{-1}x+\mu_k^T\Sigma^{-1}\mu_k$$

• $x^T\Sigma^{-1}x$ does not depend on $k$

• $x^T\Sigma^{-1}\mu_k$ is a real number, it thus equal to its transpose which is (provided $\Sigma$ is symmetric) $\mu_k^T\Sigma^{-1}x$

Thus $$(x-\mu_k)^T\Sigma^{-1}(x-\mu_k) = C - 2 x^T\Sigma^{-1}\mu_k + \mu_k^T\Sigma^{-1}\mu_k$$