 # Suppose A = QR is a QR factorization of an m xx n matrix A (with linearly independent columns). Partition A as [A_1 A_2], where A_1 has p columns. Show how to obtain a QR factorization of A_1, and explain why your factorization has the appropriate properties. vestirme4 2021-01-19 Answered
Suppose A = QR is a QR factorization of an $m×n$ matrix A (with linearly independent columns). Partition A as $\left[{A}_{1}{A}_{2}\right],where{A}_{1}$ has p columns. Show how to obtain a QR factorization of ${A}_{1}$, and explain why your factorization has the appropriate properties.
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Step 1
Given information:
Suppose A = QR is a QR factorization of an $m×n$ matrix A(with linearly independent columns).
Partition A as $\left[{A}_{1}{A}_{2}\right],where{A}_{1}$ has p columns.
Step 2
Explanation:
Write the partition Q as follows:
$Q=\left[{Q}_{1}{Q}_{2}\right],where{Q}_{1}$ has p columns.
Write the partition of R as follows:
$R=\left[\begin{array}{cc}{R}_{11}& {R}_{12}\\ 0& {R}_{22}\end{array}\right],where{R}_{11}isap×p$ matrix.
Step 3
Substitute the partition values of Q and R in the partition A.
$A=\left[{A}_{1}{A}_{2}\right]=\left[{Q}_{1}{Q}_{2}\right]\left[\begin{array}{cc}{R}_{11}& {R}_{12}\\ 0& {R}_{22}\end{array}\right]$
$=\left[{Q}_{1}{R}_{11},{Q}_{1}{R}_{12}+{Q}_{2}{R}_{22}\right]$
Refer to the calculated equation:
The value of ${A}_{1}is{A}_{1}={Q}_{1}{R}_{11}$
The matrix ${Q}_{1}$ has orthogonal columns since it is derivedfrom Q.
The matrix ${R}_{11}$ is a square and upper triangular matrix.
The diagonal entries ${R}_{11}$ are positive since these are diagonal entries of the matrix R.
Therefore, ${Q}_{1}{R}_{11}$ is a QR factorization of ${A}_{1}$.