Quadratic minimisation problems in statistics

We consider the problem subject to where  is positive definite or positive semi-definite. Variants of this problem are discussed within the framework of a general unifying methodology. These include non-trivial considerations that arise when (i)  and/or  are not of full rank and (ii)  takes special forms (especially which, under further conditions, reduces to the well-known two-sided eigenvalue solution). Special emphasis is placed on insights provided by geometrical interpretations.