the mean. Re-write the likelihood in the log form using the trace trick: The differential of this log-likelihood is It naturally breaks down into the part related to the estimation of the mean, and to the part related to the estimation of the variance. The first order condition for maximum, , is satisfied when the terms multiplying and are identically zero. Assuming (the maximum likelihood estimate of) is non-singular, the first order condition for the estimate of the mean vector is which leads to the maximum likelihood estimator This lets us simplify as defined above. Then the terms involving in can be combined as The first order condition will hold when the term in the square bracket is (matrix-valued) zero. Pre-multiplying the latter by and dividing by gives which of course coincides with the canonical derivation given earlier. Dwyer points out that decomposition into two terms such as appears above is "unnecessary" and derives the estimator in two lines of working. Note that it may be not trivial to show that such derived estimator is the unique global maximizer for likelihood function. Intrinsic covariance matrix estimation Intrinsic expectation Given a sample of n independent observations x1,..., xn of a p-dimensional zero-mean Gaussian random variable X with covariance R, the maximum likelihood estimator of R is given by The parameter belongs to the set of positive-definite matrices, which is a Riemannian manifold, not a vector space, hence the usual vector-space notions of expectation, i.e. "", and estimator bias must be generalized to manifolds to make sense of the problem of covariance matrix estimation. This can be done by defining the expectation of a manifold-valued estimator with respect to the manifold-valued point as where are the exponential map and inverse exponential map, respectively, "exp" and "log" denote the ordinary matrix exponential and matrix logarithm, and E[·] is the ordinary expectation operator defined on a vector space, in this case the tangent space of the manifold. Bias of the sample covariance matrix The intrinsic bias vector field of the SCM estimator is defined to be The intrinsic estimator bias is then given by . For complex Gaussian random variables, this bias vector field can be shown to equal where and ψ(·) is the digamma function. The intrinsic bias of the sample covariance matrix equals and the SCM is asymptotically unbiased as n → ∞. Similarly, the intrinsic inefficiency of the sample covariance matrix