# Prove that every permutation matrix is orthogonal?

Prove that every permutation matrix is orthogonal

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• Eugene
Lv 7
7 years ago
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Let P be an n x n permutation matrix. For some permutation s on {1,...,n}, P_ij = d_s(i)j, where d_kl = 1 if k = l and 0 otherwise. The (i,j) entry of PP^t is

Sum{k = 1, n} P_ik P_jk = Sum{k = 1, n} d_s(i)k d_s(j)k = d_s(i)s(j).

Since s is a permutation, s(i) = s(j) if and only if i = j. Therefore d_s(i)s(j) = d_ij. It follows that PP^t = I. By a similar argument, P^tP = I. Hence, P is orthogonal.

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