Permutation Matrix

Question

Show: If P is a permutation matrix, then P^T and P^H are also permutation matrices.

Answer 1

Suppose P is an n x n permutation matrix, and let s be the permutation associated with P. Let e₁,..., e_n and e¹,...,eⁿ be the n columns and n rows of the identity matrix I, respectively.  Then e^j P = e^s(j) for all j. Note

 

(P^T e_j)^T = (e_j)^T (P^T)^T = e^T P = e^s(j) = (e_s(j))^T  (j = 1,2,3,.., n)

 

and hence P^T e_j = e_s(j) for all j, showing that P^T is a permutation matrix (with associated permuation s^-1). 

 

Since P is a real matrix, its adjoint P^H equals P^T, so P^H is a permuration matrix.