HOMOGEN Function

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HOMOGEN Function

solves homogeneous linear systems

HOMOGEN( matrix)

where matrix is a numeric matrix or literal.

The HOMOGEN function solves the homogeneous system of linear equations A*X = 0 for X. For at least one solution vector X to exist, the m ×n matrix A, $m \geq n$ , has to be of rank r < n. The HOMOGEN function computes an n ×(n-r) column orthonormal matrix X with the property A*X = 0, X' X = I. If A'A is ill conditioned, rounding-error problems can occur in determining the correct rank of A and in determining the correct number of solutions X. Consider the following example (Wilkinson and Reinsch 1971, p. 149):

   a={22  10   2   3   7,
      14   7  10   0   8,
      -1  13  -1 -11   3,
      -3  -2  13  -2   4,
       9   8   1  -2   4,
       9   1  -7   5  -1,
       2  -6   6   5   1,
       4   5   0  -2   2};
   x=homogen(a);

These statements produce the solution

          X             5 rows      2 cols    (numeric)

                       -0.419095          0
                       0.4405091  0.4185481
                       -0.052005  0.3487901
                       0.6760591   0.244153
                       0.4129773  -0.802217

In addition, this function could be used to determine the rank of an m ×n matrix A, $m \geq n$ .

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