\newcommand{\re}{\operatorname{Re}}\newcommand{\im}{\operatorname{Im}}\newcommand{\vec}{\operatorname{vec}} \newcommand{\mat}{\operatorname{mat}} First some notation:
- Use \succeq 0 to mean matrix positive semi-definiteness.
- Use \vec as the operation that reshapes an n\times n matrix into a n^2-vector by stacking its columns: \vec := ([\mathbf{c} _1,\dots, \mathbf{c} _n] \mapsto \left[\begin{smallmatrix} \mathbf{c} _1 \\ \vdots \\ \mathbf{c} _n \end{smallmatrix} \right]), and write \mat as the backwards operation.
- For a square matrix \mathbf{M} denote the 2\times 2 minor corresponding to indices (i,j) as \mathbf{M} _{(i,j)} = \left[ \begin{smallmatrix} \mathbf{M} _{i,i} & \mathbf{M} _{i,j} \\ \mathbf{M} _{j,i} & \mathbf{M} _{j,j} \end{smallmatrix} \right].
The setup:
- \mathbf{\Sigma}\in \mathbb{C}^{n\times n} is a covariance matrix we want to estimate.
- \mathbf{A}\in \mathbb{C}^{m\times n} is known and non-degenerate with m\geq n.
- \mathbf{z} \in \mathbb{C}^m is a noise vector with zero mean and distribution f _{\mathbf{z}}
- We observe \mathbf{v} = \mathbf{A}(\vec \mathbf{\Sigma})+\mathbf{z}
We will examine \hat{\mathbf{s}} the "max-likelihood value for \vec \mathbf{\Sigma} given \mathbf{v}":
By Sylvester's criterion the constraint that (\mat \mathbf{s}) \succeq 0 is equivalent to the same constraint on each of (\mat \mathbf{s})'s 2\times 2 minors:
Consider one of these constraint subsets. The eigenvalues (u,v) for any 2\times 2 complex Hermitian matrix \mathbf{H} := \left[\begin{smallmatrix} a & b' \\ b & c\end{smallmatrix}\right] can be computed directly using trace and determinant rules:
And \mathbf{H} \succeq 0 iff u,v \geq 0, viz. (considering (\ref{eqn:uv}))
Rewriting in explicit real-quadratic form,
You can check that \mathbf{Q} is negative semi-definite. As a result the set on the left of (\ref{eqn:syl}) comprises:
- {n \choose 2} real negative semi-definite quadratic constraints from the \mathbf{Q} matrices relating the diagonals and off-diagonals,
- n linear inequality constraints enforcing that the diagonals of \mat \mathbf{s} be non-negative.
In total there are n^2 real variables involved:
- The n diagonal components of \mat{\mathbf{s}},
- The {n \choose 2} components in the upper-triangular part of \re(\mat{\mathbf{s}}),
- The {n \choose 2} components in the upper-triangular part of \im(\mat{\mathbf{s}}).
As long as the objective in (\ref{eqn:ml}),\ (\mathbf{s}\mapsto f _{\mathbf{z}}(\mathbf{v}-\mathbf{A}\mathbf{s})), considered as a function of the above n^2 real variables, is amenable to optimization then we can efficiently compute \hat{\mathbf{s}} using, say, an interior-point method. This is the case, say, when f _\mathbf{z} follows some positive-definite quadratic form on \mathbb{C}^m.