Completing Patterns to Satisfy the Anticommutativity Equation

Matrix completion problems attempt to fill in entries of a matrix so that it has interesting properties. For the anticommutativity completion problem we explore what patterns of X we can fill in to satisfy the linear matrix equation AX = - XA. In this equation the matrix X is a partial matrix with a pattern of specified and unspecified entries and the matrix A is skew symmetric; both X and A are square matrices. We utilize two different approaches to find patterns of specified entries in X that allow it to be completed to anticommute with A. One approach transforms AX = - XA into a linear equation by applying the tensor product. The other provides a means to construct and examine a basis of the nullspace of the linear transformation of the matrix equation. Using a combination of these two approaches we show what patterns of X can be completed to satisfy the anticommutativity equation. Some restrictions on the patterns of X are that X needs to have n specified entries, must be symmetric, and cannot have symmetric locations specified. The tensor product technique and the nullspace technique used here apply to many other matrix equations.