Chapter 5.3, Problem 51E

a. Consider the matrix product $Q_1=Q_{2}S$ , where both $Q_1$ and $Q_2$ arc $n\times m$ matrices with orthonormal columns. Show that S is an orthogonal matrix.Hint:Compute $Q_1^T{Q}_1={\left(Q_{2}S\right)}^T{Q}_{2}S$ . Notethat $Q_1^T{Q}_1=Q_2^T{Q}_2=I_m$ .
b. Show that the QR factorization of an $n\times m$ matrix M is unique. Hint: If $M=Q_1{R}_1=Q_2{R}_2$ , then $Q_1=Q_2{R}_2{R}_1^{-1}$ . Now use part (a) and Exercise 50a.