Miscellaneous Reverse Order Laws for Generalized Inverses of Matrix
Products with Applications
Abstract
One of the fundamental research problems in the theory of generalized
inverses of matrices is to establish reverse order laws for generalized
inverses of matrix products. Under the assumption that $A$, $B$, and
$C$ singular matrices of the appropriate sizes, two reverse order laws
for generalized inverses of the matrix products $AB$ and $ABC$ can
be written as $(AB)^{(i,\ldots,j)} =
B^{(i_2,\ldots,j_2)}A^{(i_1,\ldots,j_1)}$
and $(ABC)^{(i,\ldots,j)} =
C^{(i_3,\ldots,j_3)}B^{(i_2,\ldots,j_2)}A^{(i_1,\ldots,j_1)}$,
or other mixed reverse order laws. These equalities do not necessarily
hold for different choices of generalized inverses of the matrices. Thus
it is a tremendous work to classify and derive necessary and sufficient
conditions for the reverse order law to hold because there are all 15
types of $\{i,\ldots,
j\}$-generalized inverse for a given matrix according
to combinatoric choices of the four Penrose equations. In this paper, we
shall establish several decades of mixed reverse order laws for
$\{1\}$- and
$\{1,2\}$-generalized inverses of
$AB$ and $ABC$, and give a classified investigation to a family of
reverse order laws $(ABC)^{(i,\ldots,j)} =
C^{-1}B^{(k,\ldots,l)}A^{-1}$ for the
eight commonly-used types of generalized inverses by means of the block
matrix representation method (BMRM) and the matrix rank method (MRM). A
variety of consequences and applications these reverse order laws are
presented.