Nullspace Completions For Lyapunov Equations

In applied and theoretical mathematics, it is often desirable to complete a
matrix whose entries are partially specied in such a way that it satises a
given property. Our research focuses on completing a matrix so that is satises
a special case of the Lyapunov Equation, AX XA^T = 0. Completing a matrix
so that it satises this equation has many applications. For example, if X is
also symmetric, then X transforms a non-symmetric eigenvalue problem to a
simpler symmetric eigenvalue problem. Given a square matrix A and a partial
matrix pattern of specied and unspecied entries in a partial matrix X, we
determine when X has a completion satisfying AX -XA^T = 0. We develop
two methods using techniques from linear algebra. Ourrst method is rewriting
the equation using the Kronecker product and examining the resulting linear
equation. Our second method is constructing a basis for the nullspace of the
linear transformation corresponding to the matrix equation and using the basis
to determine which partial matrix patterns have completions.