James M. Hyman
Abstract
Large systems of linear equations often contain far fewer active degrees of freedom than their ambient dimension suggests. A particularly transparent instance occurs when a large matrix factors as A = TSW, where T is m x n, S is n x n, and W is n x m, and m >> n. The action of A then passes through an n-dimensional intermediate space. Sylvester's theorem relates the nonzero eigenvalues of a product XY to those of the reversed product YX. Applied cyclically, it shows that the nonzero spectrum of the large matrix A is completely determined by either of the small matrices B = (WT)S, C = S(WT). This article develops the theorem from first principles and explains the geometry behind the factorization. The result yields exact reduced systems, both for linear algebraic equations and for first-order linear ordinary differential equations. Worked examples carry this through in detail: spectral reduction, reconstruction of full-space eigenvectors, reduced solution of shifted systems, and exact integration of a three-dimensional ODE through a two-dimensional model. A separate section takes up singular values. Those of A, B, and C do not agree in general, though they do when the bases are orthonormal, a case the same theorem settles once it is applied to A*A. The closing sections separate exact factorization from projection-based approximation, and flag what the theorem leaves undetermined: zero-eigenvalue structure, conditioning, and transient behavior. The material is classical. What this article adds is one continuous development, from first principles through to the limits of the theorem, pitched for an upper-level undergraduate course.