A full‑rank matrix always has a closest rank‑deficient counterpart that is only one rank lower. The proof uses smooth matrix norms: for any such norm the distance to the nearest singular matrix is achieved when one singular value is set to zero, reducing the rank by one. This result applies to many ordinary distance measures, such as the Frobenius norm.

How to find that nearest matrix is straightforward when the Euclidean (Frobenius) distance is used. The optimal solution is obtained by performing a singular value decomposition (SVD) of the target matrix and then truncating it: the smallest singular value is removed (or set to zero) and the remaining singular vectors are kept. This truncation yields the matrix with minimal Frobenius distance that has the prescribed lower rank.

However, the choice of norm matters. Different unitarily invariant norms can lead to different minimizers. In a small illustrative example, the minimizers under the Frobenius, spectral and nuclear norms are distinct, with the Euclidean distance providing one particular global minimum.

There are also hard cases. Even though the weighted Frobenius norm is still induced by an inner product, finding the nearest matrix of rank ≤ 1 with respect to this norm is NP‑hard for real matrices. This shows that the ease of the problem depends critically on the specific norm and weight structure.

The main takeaway is that while nearest rank‑deficient matrices can be computed efficiently for many common norms, for other norm choices the problem can become computationally intractable.