efficient way to represent a lower/upper triangular matrix

If you have N items then a lower triangular array without the main diagonal will have (N - 1) * N / 2 elements, or (N + 1) * N / 2 elements with the main diagonal. Without the main diagonal, (I, J) (I,J ∈ 0..N-1, I > J) ⇒ (I * (I - 1) / 2 + J). With the main diagonal, (I,J ∈ 0..N-1, I ≥ J) ⇒ ((I + 1) * I / 2 + J).

(And yes, when you're allocating 4 gigabytes on a 2.5 gigabyte machine, cutting it half does make a huge difference.)

Really, you're best off just using a regular two dimensional matrix. RAM is pretty cheap. If you really don't want to do that, then you can build a one-dimensional array with the right number of elements and then figure out how to access each element. For example, if the array is structured like this:

    j
    1234
i 1 A
  2 BC
  3 DEF
  4 GHIJ

and you have it stored as a one dimensional array, left to right, you'd access element C (2, 2) with array[3]. You can work out a function to go from [i][j] to [n] but I won't spoil your fun. But you don't have to do this unless the triangular array in question is really huge or you're very concerned about space.