A vector space formulation of a ladder network synthesis problem is given and algorithms for the solution are obtained, which provide a concise and flexible synthesis method. Specifically, the problem of realizing a specified reflection coefficient with aRLCladder network is treated as a matrix synthesis problem. The matrix to be synthesized is both tridiagonal and cyclic, properties that are utilized to obtain a solution and relate it to the synthesis procedures employing continued fraction expansions. The solution is effected by tridiagonalization of a pair of matrices, a procedure observed to be a vector space analog of the Euclidean algorithm. Algorithms are given, which permit the user to specify the representation of the vectors that are formed recursively during tridiagonalization. This capability, in effect, permits the polynomials that are formed in continued fraction expansions to be represented as linear combinations of various sets of polynomials, such as the Lagrange interpolation polynomials. Experiments are described, which illustrate the advantages of the new algorithms that allow the representation to be specified.