Can someone point out the development from Eq(1) to Eq(2) of the following maximization over k ?Eq(1): − 1 2 ( x − μ k ) T Σ − 1 ( x − μ k )Eq(2): C − 1 2 μ k T Σ − 1 μ k + x T Σ − 1 μ k where C denotes a constant

Question

Can someone point out the development from Eq(1) to Eq(2) of the following maximization over   k ?Eq(1):   −      1    2    (  x  −      μ    k        )    T        Σ          −      1        (  x  −      μ    k    )Eq(2):   C  −      1    2        μ    k    T        Σ          −      1            μ    k    +      x    T        Σ          −      1            μ    k   where   C denotes a constant

lisicw2 · Accepted Answer

(  x  −      μ    k        )    T        Σ          −      1        (  x  −      μ    k    )  =      x    T        Σ          −      1        x  −      x    T        Σ          −      1            μ    k    −      μ    k    T        Σ          −      1        x  +      μ    k    T        Σ          −      1            μ    k  1.       x    T        Σ          −      1        x does not depend on   k2.       x    T        Σ          −      1            μ    k   is a real number, it thus equal to its transpose which is (provided   Σ is symmetric)       μ    k    T        Σ          −      1        xThus  (  x  −      μ    k        )    T        Σ          −      1        (  x  −      μ    k    )  =  C  −  2      x    T        Σ          −      1            μ    k    +      μ    k    T        Σ          −      1            μ    k

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

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Answer & Explanation

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