GINV Function :: SAS/IML(R) 13.1 User's Guide

GINV Function

GINV (matrix) ;

The GINV function computes the Moore-Penrose generalized inverse of matrix. This inverse, known as the four-condition inverse, has these properties:

If $\mb {G} = \mbox{GINV}(\mb {A})$ then

$\mb {AGA} = \mb {A} ~ ~ ~ \mb {GAG}= \mb {G} ~ ~ ~ (\mb {AG})^{\prime } = \mb {AG} ~ ~ ~ (\mb {GA})^{\prime } = \mb {GA} ~$

The generalized inverse is also known as the pseudoinverse, usually denoted by $\mb {A}^{-}$ . It is computed by using the singular value decomposition (Wilkinson and Reinsch, 1971).

See Rao and Mitra (1971) for a discussion of properties of this function.

As an example, consider the following model:

$\mb {Y}= \mb {X} \bbeta + \bepsilon$

Least squares regression for this model can be performed by using the quantity ginv(x)*y as the estimate of $\bbeta$ . This solution has minimum $\mb {b}^{\prime }\mb {b}$ among all solutions that minimize $\bepsilon ^{\prime }\bepsilon$ , where $\bepsilon = \mb {Y} - \mb {Xb}$ .

Projection matrices can be formed by specifying GINV $(\mb {X})*\mb {X}$ (row space) or $\mb {X}*$ GINV $(\mb {X})$ (column space).

The following program demonstrates some common uses of the GINV function:

A = {1 0 1 0 0,
     1 0 0 1 0,
     1 0 0 0 1,
     0 1 1 0 0,
     0 1 0 1 0,
     0 1 0 0 1 };

/* find generalized inverse */
Ainv = ginv(A);

/* find LS solution: min  |Ax-b|^2 */
b = { 3, 2, 4, 2, 1, 3 };
x = Ainv*b;

/* form projection matrix onto row space.
   Note P = P` and P*P = P */
P = Ainv*A;

/* find numerical rank of A */
rankA = round(trace(P));

reset fuzz;
print Ainv, rankA, x, P;

Figure 24.144: Common Uses of the Generized Inverse

Ainv
0.2666667	0.2666667	0.2666667	-0.066667	-0.066667	-0.066667
-0.066667	-0.066667	-0.066667	0.2666667	0.2666667	0.2666667
0.4	-0.1	-0.1	0.4	-0.1	-0.1
-0.1	0.4	-0.1	-0.1	0.4	-0.1
-0.1	-0.1	0.4	-0.1	-0.1	0.4

rankA
4

x
2
1
1
0
2

P
0.8	-0.2	0.2	0.2	0.2
-0.2	0.8	0.2	0.2	0.2
0.2	0.2	0.8	-0.2	-0.2
0.2	0.2	-0.2	0.8	-0.2
0.2	0.2	-0.2	-0.2	0.8

If $\mb {A}$ is an $n\times m$ matrix, then, in addition to the memory allocated for the return matrix, the GINV function temporarily allocates an $n^2 + nm$ array for performing its computation.