Question

a.Consider the matrix product ${\mathit{Q}}_{\mathbf{1}}\mathbf{=}{\mathit{Q}}_{\mathbf{2}}\mathit{S}$, where both ${\mathit{Q}}_{\mathbf{1}}$and ${\mathit{Q}}_{\mathbf{2}}$ are n×mmatrices with orthonormal columns. Show that Sis an orthogonal matrix. Hint: Computelocalid="1659499054761" ${{\mathbf{Q}}_{\mathbf{1}}}^{\mathbf{T}}\mathbf{=}{\mathit{Q}}_{\mathbf{1}}\mathbf{=}{\left(Q_{2}S\right)}^{\mathbf{T}}{\mathit{Q}}_{\mathbf{2}}\mathit{S}$. Note that

${{\mathit{Q}}_{\mathbf{1}}}^{\mathbf{T}}\mathbf{=}\mathit{Q}\mathbf{=}{{\mathbf{Q}}_{\mathbf{2}}}^{\mathbf{T}}\mathbf{=}{\mathit{Q}}_{\mathbf{2}}\mathbf{=}{\mathit{I}}_{\mathbf{m}}$

b.Show that the QRfactorization of an n×mmatrix Mis unique. Hint: If${\mathbf{Q}}_{\mathbf{1}}{\mathit{R}}_{\mathbf{1}}\mathbf{=}{\mathbf{Q}}_{\mathbf{2}}{\mathit{R}}_{\mathbf{2}}$, then${\mathbf{Q}}_{\mathbf{1}}\mathbf{=}{\mathbf{Q}}_{\mathbf{2}}{\mathbf{R}}_{\mathbf{2}}{{\mathit{R}}_{\mathbf{1}}}^{\mathbf{-}\mathbf{1}}$ . Now use part (a) and Exercise 50a.

Short answer

1. Matrix S is orthogonal.
2. QR factorization of any matrix is unique.

Step-by-step solution

S is orthogonal.

(a)

Observe the following terms

$Q_1=Q_{2}S\Rightarrow S={Q_2}^T{Q}_1$

In order to show that S is orthogonal, we need to show that $S^{T}S=l_m$.Consider

$$\begin{gathered} S^{T}S={\left({Q_2}^T{Q}_1\right)}^T\left({Q_2}^T{Q}_1\right) \\ =\left({Q_1}^T{\left({Q_2}^T\right)}^T\right)\left({Q_2}^T{Q}_1\right) \\ ={Q_1}^T\left(Q_2{Q_2}^T\right)Q_1 \\ =Q_1{I}_n{Q}_1 \\ =I_m \end{gathered}$$

Thus, S is orthogonal.

Consider for part (b).

b)

Let M an n× mmatrix and let $M=Q_1{R}_1=Q_{2}R$ be QR factorization of M.

Also, recall that the diagonals entries of the matrices$R_1$and$R_2$are strictly positive. Now, it is written as,

$$\begin{gathered} Q_1{R}_1=Q_2{R}_2 \\ =Q_1=Q_2{R_2}^{-1} \end{gathered}$$

From part of this series, $R_1{R_1}^{-1}$is an orthogonal matrix. So, it is found as,

$$\begin{gathered} R_2{R_1}^{-1}=I \\ \Rightarrow R_1=R_2 \end{gathered}$$

And it gives $Q_1=Q_2$.

Hence,

1. Matrix S is orthogonal matrix.
2. QR factorization of any matrix is unique.