21.1.2 Creating Permutation Matrices

For creating permutation matrices, Octave does not introduce a new function, but rather overrides an existing syntax: permutation matrices can be conveniently created by indexing an identity matrix by permutation vectors. That is, if q is a permutation vector of length n, the expression

       P = eye (n) (:, q);

will create a permutation matrix - a special matrix object.

     eye (n) (q, :)

will also work (and create a row permutation matrix), as well as

     eye (n) (q1, q2).

For example:

       eye (4) ([1,3,2,4],:)
     ⇒
     Permutation Matrix
     
        1   0   0   0
        0   0   1   0
        0   1   0   0
        0   0   0   1
     
       eye (4) (:,[1,3,2,4])
     ⇒
     Permutation Matrix
     
        1   0   0   0
        0   0   1   0
        0   1   0   0
        0   0   0   1

Mathematically, an identity matrix is both diagonal and permutation matrix. In Octave, eye (n) returns a diagonal matrix, because a matrix can only have one class. You can convert this diagonal matrix to a permutation matrix by indexing it by an identity permutation, as shown below. This is a special property of the identity matrix; indexing other diagonal matrices generally produces a full matrix.

       eye (3)
     ⇒
     Diagonal Matrix
     
        1   0   0
        0   1   0
        0   0   1
     
       eye(3)(1:3,:)
     ⇒
     Permutation Matrix
     
        1   0   0
        0   1   0
        0   0   1

Some other built-in functions can also return permutation matrices. Examples include inv or lu.