1 Introduction In the matrix completion problem we are given a partial symmetric real ma 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 Collections: Mathematics