312 CHAPTER 12. SELF ADJOINT OPERATORS

am following a nice discussion given in Wikipedia which makes use of the existence of squareroots.

What people do to estimate these is to take n independent observations x1, · · · ,xn andtry to predict what m and Σ should be based on these observations. One criterion used formaking this determination is the method of maximum likelihood. In this method, you seekto choose the two parameters in such a way as to maximize the likelihood which is given as

n∏i=1

1

det (Σ)1/2

exp

(−1

2(xi−m)

∗Σ−1 (xi−m)

).

For convenience the term (2π)p/2

was ignored. Maximizing the above is equivalent to max-imizing the ln of the above. So taking ln,

n

2ln(det(Σ−1

))− 1

2

n∑i=1

(xi−m)∗Σ−1 (xi−m)

Note that the above is a function of the entries of m. Take the partial derivative withrespect to ml. Since the matrix Σ−1 is symmetric this implies

n∑i=1

∑r

(xir −mr) Σ−1rl = 0 each l.

Written in terms of vectors,n∑

i=1

(xi −m)∗Σ−1 = 0

and so, multiplying by Σ on the right and then taking adjoints, this yields

n∑i=1

(xi −m) = 0, nm =

n∑i=1

xi, m =1

n

n∑i=1

xi ≡ x̄.

Now that m is determined, it remains to find the best estimate for Σ. (xi−m)∗Σ−1 (xi−m)

is a scalar, so since trace (AB) = trace (BA) ,

(xi−m)∗Σ−1 (xi−m) = trace

((xi−m)

∗Σ−1 (xi−m)

)= trace

((xi−m) (xi−m)

∗Σ−1

)Therefore, the thing to maximize is

n ln(det(Σ−1

))−

n∑i=1

trace((xi−m) (xi−m)

∗Σ−1

)

= n ln(det(Σ−1

))− trace

S︷ ︸︸ ︷(

n∑i=1

(xi−m) (xi−m)∗

)Σ−1

We assume that S has rank p. Thus it is a self adjoint matrix which has all positive eigen-values. Therefore, from the property of the trace, the thing to maximize is

n ln(det(Σ−1

))− trace

(S1/2Σ−1S1/2

)