(7) (AIA) Ly] =(0 
E21 
nxm mxp nxp 
where E» is a (pxp)-unit matrix. 
By definition D -L 
21*. 1 721 
E21 
As shown in (7) matrix Dy, is the null-space (or kernel) of the singular matrix A. 
Assuming that Q, is regular the null-space of the singular normal matrix N = 
(Atpa) equals the null-space of A, hence the rank-deficiency of N is p. 
The least-squares solution (4) now can be rewritten as: 
(8) N.AX s Ab 
where 
P 
X 
" 
x 
| 
X 
= AL AL 
ls 
Since N is singular there are more solutions with AX fulfilling equation (8). 
(8) is called an inhomogenious set of normal equations. 
An homogenious set of normal equations is obtained by setting 
(9) N.u - (0) 
-7398:-