 
Summary: 1 Introduction
In the matrix completion problem we are given a partial symmetric real ma
trix A with certain elements specified or fixed and the rest are unspecified or
free; and, we are asked whether A can be completed to satisfy a given prop
erty (P ) by assigning certain values to its free elements. In this chapter, we
are interested in the following two completion problems: the positive semidef
inite matrix completion problem corresponding to (P) being the positive semi
definiteness (PSD) property; and the Euclidean distance matrix completion
problem corresponding to (P) being the Euclidean distance (ED) property.
(We concentrate on the latter problem. A survey of the former is given in [?].
The relationships between the two is discussed in e.g. [?, ?, ?].) These prob
lems can be generalized to allow for approximate completion by formulating
the completion problems as weighted closest matrix problems:
min ff(X) = jj H ffi (A \Gamma X) jj 2
F : for all X satisfying (P)g; (1)
where jj : jj F denotes the Frobenius norm defined as jj A jj F =
p
trace A T A,
(ffi) denotes the Hadamard or the elementwise product, and the weight matrix
H is symmetric with nonnegative elements. If each element of the weight
