APPLIED MULTIVARIATE STATISTICS: WITH SAS® SOFTWARE

A.12. Generalized Inverse of a Matrix

For any matrix say B, one can find a matrix say G such that BGB = B. The matrix G is called a generalized inverse (or g-inverse for short) of B. A g-inverse always exists but is not unique unless the matrix B is square and nonsingular (in which case, the regular inverse is the unique g-inverse).

SAS produces one particular g-inverse called the Moore Penrose inverse which in addition to above requirement satisfies some more conditions (Rao, 1973). It is obtained by the statement,

g = ginv(b);

For the 2 by 3 matrix B defined as,

the Moore Penrose g-inverse is obtained as a 3 by 2 matrix,