An enormous share of inverse problems are linear: the data depends linearly on the model. Then the forward map is a matrix, and the whole problem is the system

with m \in \mathbb{R}^n the unknown model, d \in \mathbb{R}^M the data, and G the M \times n forward operator. Each row of G is one measurement — a recipe that weights the model parameters and reports a number. Inverting the problem means solving this system for m.

Three shapes of problem

The shape of G already tells you what to expect:

• Even-determined (M = n, G invertible): a unique solution m = G^{-1}d — the rare, easy case.
• Over-determined (M > n): more measurements than unknowns. Usually no exact solution because the data is noisy and inconsistent — we seek a best fit (least squares).
• Under-determined (M < n): fewer measurements than unknowns. Infinitely many models fit — we must choose among them (minimum norm, or a prior).

Real problems are often both at once — over-determined in some directions, under-determined in others — which is exactly what the SVD is built to untangle.