The Minimal Rank of the Matrix Expression $A - BX - YC$

The minimal rank of the matrix expression A BX Y C with respect to the choice of X and Y are determined using generalized inverses of matrices. Some of their applications are also presented. Suppose that p(X; Y ) = A BX Y C (1) is a linear matrix expression over the complex number field, where A, B , and C are m n, m k, and l n matrices, respectively; X and Y are k n and m l variant matrices, respectively. In this article we consider the minimal rank of p(X; Y ) with respect to the choice of X and Y , and present some of their applications. To do so, we need some well-known formulas related to ranks and generalized inverse of matrices. Lemma 1 [2] [3]. Let A 2 Cm×n, B 2 Cm×k and C 2 Cl×n be given. Then they satisfy the rank equalities r [A; B ] = r (A) + r (B AA −B ) = r (B ) + r (A BB −A); (2) r [ A C ] = r (A) + r (C CA−A) = r (C) + r (A AC −C); (3) r [ A B C 0 ] = r (B ) + r (C) + r [(I m BB −)A(I n C−C)]; (4) where ( )− denotes an inner inverse of a matrix. We are ready to establish the main result of this article. Theorem 2. The minimal rank of p(X; Y ) in (1) with respect to the choice of X and Y is min X,Y r (A BX Y C) = r [ A B C 0 ] r (B ) r (C): (5) VOLUME 14, NUMBER 1, WINTER 2002 41 The matrices X and Y satisfying (5) are given by X = B −A + UC + (I k B −B )U1; (6) Y = (I m BB −)AC − BU + U2(I l CC−); (7) where U, U1 and U2 are arbitrary. Proof. Let