0 10 20 30 40 50 60 linear combination number 10 -2 10 0 10 2 10 4 eigenvalue excited states ground states FIG. 1: (Color online)The spectra of eigenvalues from a singular-value decomposition of the sen- sitivity matrix A (Eq. 5) for the two-body random ensemble. The lower curve is for ground state energies, while the upper curve is for excitation energies. This equation is in the form ⃗ b = A ⃗ x where A$_{ij}$ = ∑ α ∂E$_{α}$ ∂v$_{i}$ ∂E$_{α}$ ∂v$_{j}$ . (5) ∂E$_{α}$ ∂v$_{i}$ = 〈 Ψ$_{α}$ ∣ ˆ H$_{i}$ ∣ Ψ$_{α}$ 〉 (6) The derivatives come via the Hellmann-Feynman theorem [20] where ˆ H$_{i}$ is the Hamitonian operator whose strength is v$_{i}$ . We then find the eigenvalues of A (which is equivalent to finding the squares of the eigenvalues in the singular-value decomposition of ∂E$_{α}$/∂v$_{i}$ ). We do this for both the USDB interaction and for an ensemble of 100 sets of random two-body interactions, also called the two-body random ensemble (TBRE). The results are shown for the TBRE in Fig. 1, where we have separated out the sensitivity just for the binding energies (ground state energies) and the excitation energies. Although not shown, the equivalent SVD eigenvalues for USDB are completely within the TBRE results. The lower curve is for ground states only, while the upper curve is for excitations energies relative to the ground state. Clearly, and perhaps unsurprisingly, the ground state energies are predominantly sensitive to just a few linear combinations of matrix elements–significantly fewer than excitations energies. Fig. 1 characterizes eigenvalues of A . The next step is to characterize the eigenvectors associated with the dominant eigenvalues, specifically by comparing with monopole and 4