Low-rank matrix approximations over canonical subspaces

In this paper we derive closed form expressions for the nearest rank-\(k\) matrix on canonical subspaces.    We start by studying three kinds of subspaces.  Let \(X\) and \(Y\) be a pair of given matrices. The first subspace contains all the \(m\times n\) matrices \(A\) that satisfy \(AX=O\). The second subspace contains all the \(m \times n\) matrices \(A\) that satisfy \(Y^TA = O\),  while the matrices in the third subspace satisfy both \(AX =O\) and \(Y^TA = 0\).   The second part of the paper considers a subspace that contains all the symmetric matrices \(S\) that satisfy \(SX =O\).  In this case, in addition to the nearest rank-\(k\) matrix we also provide the nearest rank-\(k\) positive  approximant on that subspace.   A further insight is gained by showing that the related cones of positive semidefinite matrices, and  negative semidefinite matrices, constitute a polar decomposition of this subspace. The paper ends with two examples of applications.  The first one regards the problem of computing the nearest rank-\(k\) centered matrix, and adds new insight into the PCA of a matrix. The second application comes from the field of Euclidean distance matrices.  The new results on low-rank positive approximants are used to derive an explicit expression for the nearest source matrix.  This opens a direct way for computing the related positions matrix.