Positive semidefinite solution to matrix completion problem and matrix approximation problem

Abstract

In this paper, firstly, we discuss the following matrix completion
problem in the spectral norm:
$$\left\| {\left( {\begin{array}{*{20}c}
A & B \\
{B^*} & X \\
\end{array} } \right)} \right\|_2 < 1 \quad{\mbox{\rm subject to\ }}\quad \left( {\begin{array}{*{20}c}
A & B \\
{B^*} & X \\
\end{array} } \right) \geqslant 0.$$
The feasible condition for the above problem is established, in this case, the general positive semidefinite solution and its minimum rank are presented. Secondly, applying the result of the above problem, we also study the matrix approximation problem:
$$\left\| {A-BXB^*} \right\|_2 < 1 \quad{\mbox{\rm subject to\ }}\quad A-BXB^* \geqslant 0,$$
where $A \in \mathbb{C}_\geqslant^{m \times m}$, $B \in \mathbb{C}^{m \times n}$, and $X \in \mathbb{C}_\geqslant ^{n \times n}$.