Why does the covariance matrix reduce to a scalar in this equation?

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from Machine Learning: A Probabilisic Perspective page 100 in this version.

MLAPP equation 4.15

How do we get from the far right hand side of 4.15 to 4.16? I don't see why the inverse of the covariance matrix can be reduced to become $\frac{1}{N}$...

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They are solving the equation for a value of $\mu$ such that the derivative of the likelihood function with respect to $\mu$ is 0. The fact that $\Sigma$ is invertible means you can multiply on both sides by it. It isn't a scalar.