Prove Theorem 9.10. Theorem 9.10 Suppose that Q is an orthogonal n n matrix. Then (i)

Question:

Prove Theorem 9.10.
Theorem 9.10
Suppose that Q is an orthogonal n × n matrix. Then
(i) Q is invertible with Q−1 = Qt;
(ii) For any x and y in Rn, (Qx)t Qy = xty;
In addition, the converse of part (i) holds. That is,
• any invertible matrix Q with Q−1 = Qt is orthogonal.
As an example, the permutation matrices discussed in Section 6.5 have this property, so they are orthogonal.