The relationship between the linear matrix inequality (LMI), generalized X-Y functions, and triangular factorization is examined within the framework of the classical linear-quadratic-gaussian problem. It is shown that the generalized X-Y functions arise naturally as components within the factors of the matrix forming the LMI when that matrix is decomposed into its symmetric triangular factors. This viewpoint enables us to propose a low-dimensional computational algorithm for time-dependent problems which reduces to the generalized X-Y situation for constant systems. 

In addition to the basic factorization results, we also briefly touch upon several related topics including the infinite-interval (regulator) problem, singular control problems, canonical forms, and numerical considerations.