Permutations and Permutation Matrices

1. Besides Permutation Cycles and Permutation Tables, any Permutation can also be represented using Permutation Matrices.
2. Any given Permutation (represented by a given set of Permutation Cycles or a Permutation Table) can be represented either by a Single Permutation Matrix or 2 Distinct Permutation Matrices based on the following
   1. If the Permutation consists of One or more Disjoint Permutation Cycles of Length > 2, then the Permutation can be represented by 2 Distinct Permutation Matrices (which are Transposes of each other), one that Alters the Order of Rows of any other Matrix as per the given Permutation (also known as Row Permutation Matrix) and the other that Alters the Order of Columns of any other Matrix as per the given Permutation (also known as Column Permutation Matrix)
   2. If the Permutation consists only of One or more Disjoint Permutation Cycles of Length 2, then the Permutation can be represented by a Single Permutation Matrix which Alters the Order of Rows or Columns of any other Matrix as per the given Permutation. In this case both the Row Permutation Matrix and the Column Permutation Matrix are same.
3. Conversely, any given Permutation Matrix can be represented either by a Single Permutation or 2 Distinct Permutations based on the following
   1. If for a Permutation Matrix \(P\), \(P^T \neq P\), then its Altered Order of Rows is different than its Altered Order of Columns and the Permutation Matrix can be represented by 2 Distinct Permutations, one corresponding to its Altered Order of Rows (also known as Row Permutation) and the other corresponding to its Altered Order of Columns (also known as Column Permutation). These Permutations always consists of One or more Disjoint Permutation Cycles of Length > 2.
      
      Also since for any Permutation Matrix \(P^{-1}=P^T\) (since it is an Orthogonal Matrix), the Row Permutation and Column Permutation are Inverses of each other.
   2. If for a Permutation Matrix \(P\), \(P^T = P\), then its Altered Order of Rows is same as its Altered Order of Columns and the Permutation Matrix can be represented by a Single Permutation which corresponds to both its Altered Order of Rows and Columns. In this case, the Single Permutation represents both the Row Permutation and the Column Permutation. This Permutation always consists only of One or more Disjoint Permutation Cycles of Length 2.
      
      Also since for any Permutation Matrix \(P^{-1}=P^T\) (since it is an Orthogonal Matrix), the Permutation is Inverse of itself.