Language Reference: GINV Function

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 $\text{[math]}$ then

$\text{[math]}$

The generalized inverse is also known as the pseudoinverse, usually denoted by $\text{[math]}$ . 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:

$\text{[math]}$

Least squares regression for this model can be performed by using the quantity ginv(x)*y as the estimate of $\text{[math]}$ . This solution has minimum $\text{[math]}$ among all solutions that minimize $\text{[math]}$ , where $\text{[math]}$ .

Projection matrices can be formed by specifying GINV $\text{[math]}$ (row space) or $\text{[math]}$ GINV $\text{[math]}$ (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 23.123 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 $\text{[math]}$ is an $\text{[math]}$ matrix, then, in addition to the memory allocated for the return matrix, the GINV function temporarily allocates an $\text{[math]}$ array for performing its computation.