Question

Consider a block matrix $\mathit{A}\mathbf{=}\left[\begin{array}{cc}{A}_1,& A_2\end{array}\right]$ with linearly independent columns. (A1 is an $\mathit{n}\mathbf{\times }{\mathit{m}}_{\mathbf{1}}$ matrix, and A2 is $\mathit{n}\mathbf{\times }{\mathit{m}}_{\mathbf{2}}$.) Suppose you know the QR factorization of A. Explain how this allows you to find the QRfactorization of ${\mathit{A}}_{\mathbf{1}}$.

Short answer

QR factorization of $A_1\mathrm{is}{\mathrm{Q}}_1{\mathrm{R}}_1$.

Step-by-step solution

Consider for solution.

Given is $A=\left[\begin{array}{cc}{A}_1,& A_2\end{array}\right]\mathrm{where}{\mathrm{A}}_1\mathrm{is}\mathrm{an}n\times m_1\mathrm{and}{A}_2\mathrm{is}\mathrm{an}n\times m_2$matrix.

Observe the following.

A=QR

$=\left[\begin{array}{cc}{Q}_1& Q_2\end{array}\right]\left[\begin{array}{cc}{R}_1& R_2\\ 0& R_3\end{array}\right]$

In the above expression $Q_1$and $Q_2$are two matrices of size $n\times m_1\mathrm{and}\mathrm{n}\times {\mathrm{m}}_2$where the columns are orthonormal, and $R_1,R_2,R_3$are matrices in which $R_1$and $R_3$are upper triangular matrix and as non-negative diagonal entries. Thus,

$$\begin{gathered} \left[\begin{array}{cc}{A}_1& A_2\end{array}\right]=QR \\ =\left[\begin{array}{cc}{Q}_1& Q_2\end{array}\right]\left[\begin{array}{cc}{R}_1& R_2\\ 0& R_3\end{array}\right] \\ =\left[Q_1{R}_1{Q}_1{R}_2+Q_2{R}_3\right] \end{gathered}$$

Hence, the QR factorization of $A_1$is $Q_1{R}_1$.