1 2 3 4 5 6 linear combination number 0.01 0.10 1.00 eigenvalue all states usdb random FIG. 2: (Color online) The SVD spectrum that measures the overlap of the subspace defined by the six largest eigenvectors from Fig. 1, with the subspace defined by the monopole-monopole interaction. (Black) squares are for the TBRE, while (red) diamonds are for USDB. contact interactions. To do so we first discuss a method for quantifying the overlap between two vector subspaces [21, 22]. Consider two vectors subspaces, S$_{1}$ and S$_{2}$ . Let V$_{1}$ be a matrix whose column vectors are the (orthonormal) basis vectors of S$_{1}$ , and similarly with V$_{2}$ . From these one constructs the overlap matrix Ω = V † $_{1}$V$_{2}$ . Note that if the subspaces are not of equal dimension then Ω is not a square matrix. In any case we do a singular value decomposition of Ω ; the SVD eigenvalue spectrum then is a measure of the overlap of the two spaces. If the two spaces perfectly overlap then all eigenvalues are 1, if just N of the dimensions perfectly overlap than N eigenvalues will be 1 and the rest zero. Note that this method is invariant under arbitrary choice of orthonormal bases. We begin with the monopole-monopole interaction of the form n$_{a}$ ( n$_{b}$ − δ$_{ab}$ ), which has six unique terms, and thus six vectors or linear combinations of matrix elements, in the sd -shell. These we combine with the k most dominant linear combinations that arise from the previous analysis; somewhat arbitrarily we chose k = 6 (our results do not change qualitatively for other small values of k ). The results, the SVD eigenvalues of Ω, are shown in Fig. 2. It is important to note we compute the spectrum separately for each randomly generated interaction and then compute the distribution. The results for ground states and for excited states are similar, so we combine all states into a single calculation. The eigenvalues for USDB are roughly 50% higher than for the TBRE, but otherwise qualitatively very similar. 5