Let P be an n x n matrix Prove that if P1 P then the row ve

Let P be an n x n matrix. Prove that if P^-1 = P then the row vectors of P form an Orthonormal basis for R^n.

Solution

Let P be an orthogonal matrix. Then

        P^-1 = P^T       

Here   P is an nxn matrix

Proof

We need to show that if P is orthogonal, then

        P^TP = I

This follows immediately from the definition of orthogonal and matrix multiplication. If v_j is the j^th column of P, then

        [P^TP]_ij = v_i^. v_j

But since {v₁, ..., v_n} is an orthonormal set of vectors, we have

         v_i^. v_j = d_ij

hence it is true