The following five chapters exhibit further contributions of the theory of time-variant and quasi-separable systems to matrix algebra. This chapter treats LU factorization, or, equivalently, spectral factorization, which is another, often occurring type of factorization of a quasi-separable system. This type of factorization does not necessarily exist and, when it exists, could traditionally not be computed in a stable numerical way (Gaussian elimination). Here we present necessary and sufficient existence conditions and a stable numerical algorithm to compute the factorization using orthogonal transformations applied to the quasi-separable representation.