We present a number of systolic arrays for decomposing a matrix into its lower and upper triangular factors (LU-decomposition). These architectures have been formally derived using techniques for synthesizing systolic arrays from affine recurrence equations, and the entire design process can be automated. The derivation highlights two important aspects of our synthesis methodology. Firstly, the initial specification is a high level one similar to a nested loop program. We illustrate the use a technique called explicit pipelining to automatically localize the data dependencies. Secondly, the architectures that we present have interesting features such as control signals, and specialized behavior of certain processors (such as boundary processors). We describe how these characteristics (and also processor initialization signals) can be derived automatically.