Factoring matrices into the product of two matrices

Abstract

Linear algebra of factoring a matrix into the product of two matrices with special properties is developed. This is accomplished in terms of the so-called inverse of a matrix subspace which yields an extended notion for the invertibility of a matrix. The product of two matrix subspaces gives rise to a natural generalization of the concept of matrix subspace. Extensions of these ideas are outlined. Several examples on factoring are presented. (2000): 15A23, 65F30. AMS subject classificationKey words: matrix factorization, inverse of a matrix subspace, product of matrix subspaces.

Differential geometry of matrix inversion

“…With k = 1 we are dealing with the familiar notion of nonsingularity of a matrix. If V is nonsingular, then its subset consisting of invertible V ∈ V is open and dense [12].…”

“…(For k = n(n − 1), simply take a matrix subspace with a zero row or column.) Typically matrix subspaces appearing in factorization problems satisfy k = O(n) [12].…”

“…Invertible matrix subspaces are of central relevance for solving a certain type of matrix factorization problems [12]. On the other hand it is a challenge to give a concrete characterization of Inv(V ) [13].…”

“…Theorem 3.5 can be stated locally, in terms of the quadratic function (12), yielding probably the simplest way of checking the invertibility of a nonsingular matrix subspace V . Here it is of use to observe that if V does not contain the scalars, then one can consider the equivalent matrix subspace V V −1 for an invertible V ∈ V .…”

“…1 There exists an analogous, albeit far more geometrical notion for the Grassmannian manifolds of square matrices [12], [13]. To this end, let V be a matrix subspace of C n×n over C (or R) possessing invertible elements.…”