ABSTRACT. This paper establishes that every positive-definite matrix can be written as a positive linear combination of outer products of integer-valued vectors whose entries are bounded by the geometric mean of the condition number and the dimension of the matrix. This paper addresses a geometric question that arises in the theory of discrete normal approximation [BLX15] and in the analysis of hardware for implementing matrix multiplication [LUW15] . The problem requires us to represent a nonsingular covariance matrix as a positive linear combination of outer products of integer vectors. The theoretical challenge is to obtain an optimal bound on the magnitude of the integers required as a function of the condition number of the matrix. We establish the following result. This result has an alternative interpretation as a matrix factorization: In this expression, Z is a d × r integer matrix with entries bounded by m. The r × r matrix ∆ is nonnegative and diagonal. The proof of Theorem 1.1 appears in Section 3. Section 4 demonstrates that the dependence on the condition number cannot be improved. We believe that the dependence on the dimension is also optimal, but we did not find an example that confirms this surmise. This section contains brief preliminaries. The books [HJ90, Bha97, Bar02, BV04] are good foundational references for the techniques in this paper. We use lowercase italic letters, such as c, for scalars. Lowercase boldface letters, such as z, denote vectors. Uppercase boldface letters, such as A, refer to matrices. We write z i for the i th component of a vector z, and a i j for the (i , j ) component of a matrix A. The j th column of the matrix A will be denoted by a j . We work primarily in the real linear space H d of real d × d symmetric matrices, equipped with the usual componentwise addition and scalar multiplication:
REF proved that every positive-definite matrix B admits an exact semidiscrete factorization (8.2) where V = W and the factor V has integer entries that are bounded in terms of the condition number and the dimension.
Integer factorization of a positive-definite matrix
SIAM J. Discrete Math.
SIAM J. Discrete Math.
Mathematics Computer Science