Content deleted Content added
merge sections |
→Properties: Generalizations |
||
Line 24:
== Properties ==
* If a nonsingular matrix and its inverse are both [[nonnegative matrices]] (i.e. matrices with nonnegative entries), then the matrix is a generalized permutation matrix.
== Generalizations ==
One can generalize further by allowing the entries to lie in a ring, rather than in a field. In that case if the non-zero entries are required to be [[units of a ring|units]] in the ring (invertible), one again obtains a group. On the other hand, if the non-zero entries are only required to be non-zero, but not necessarily invertible, this set of matrices forms a [[semigroup]] instead.
One may also schematically allow the non-zero entries to lie in a group ''G,'' with the understanding that matrix multiplication will only involve multiplying a single pair of group elements, not "adding" group elements. This is an [[abuse of notation]], since element of matrices being multiplied must allow multiplication and addition, but is suggestive notion for the (formally correct) abstract group <math>G \wr S_n</math> (the wreath product of the group ''G'' by the symmetric group).
==Signed permutation group==
| |||