The Sparse Basis Problem and Multilinear Algebra

Let A be a k by n underdetermined matrix The sparse basis problem for the row space W of A is to nd a basis of W with the fewest number of nonzeros Suppose that all the entries of A are nonzero and that they are algebraically independent over the rational number eld Then every nonzero vector in W has at least n k nonzero entries Those vectors in W with exactly n k nonzero entries are the elementary vectors of W A simple combinatorial condition that is both necessary and su cient for a set of k elementary vectors of W to form a basis of W is presented here A similar result holds for the null space of A where the elementary vectors now have exactly k nonzero entries These results follow from a theorem about nonzero minors of order m of the m st compound of an m by n matrix with algebraically independent entries which is proved using multilinear algebra techniques This combinatorial condition for linear independence is a rst step towards the design of algorithms that compute sparse bases for the row and null space without imposing arti cial structure constraints to ensure linear independence AMS MOS subject classi cations primary F K A