Approximation by matrices positive semidefinite on a subspace
The authors study the following problem: Let \(A\in {\mathbb{R}}^{n\times n}\), \(S\in {\mathbb{R}}^{n\times p}\) \((p<n)\), rank S\(=p\), minimize \(\| A- P\|\), subject to \(P=P^ T\), \(x^ TPx\geq 0\) \(\forall x:\) \(S^ Tx=0\). Here \(\| A\|\) is either the Frobenius norm \((\| A\| =(\sum | a_{ij}|^ 2)^{1/2})\) or the 2-norm \((\| A\|_ 2=\sup (\| Ax\| | \quad \| x\| =1),\quad \| x\| =(\sum | x_ i|^ 2)^{1/2}).\) The authors describe the (unique) minimizing matrix in the case of the Frobenius norm and a minimizing matrix in the case of the 2-norm when A is symmetric. The results are applied to give two new characterizations for Euclidean distance matrices.
scientific article; zbMATH DE number 4091674