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

vestirme4

Answered question

2021-01-19

Suppose A = QR is a QR factorization of an m×n matrix A (with linearly independent columns). Partition A as [A1A2],whereA1 has p columns. Show how to obtain a QR factorization of A1, and explain why your factorization has the appropriate properties.

Answer & Explanation

Aamina Herring

Aamina Herring

Skilled2021-01-20Added 85 answers

Step 1 
Given information: 
Suppose A = QR is a QR factorization of an m×n matrix A(with linearly independent columns). 
Partition A as [A1A2],whereA1 has p columns. 
Step 2 
Explanation: 
Partition Q should be written as follows:
Q=[Q1Q2],whereQ1 has p columns. 
As you write R's partition, you should:
R=[R11R120R22],whereR11isap×p matrix. 
Step 3 
Replace Q and R's partition values in the partition A.
A=[A1A2]=[Q1Q2][R11R120R22] 
=[Q1R11,Q1R12+Q2R22] 
Refer to the calculated equation: 
The value of A1isA1=Q1R11 
The matrix Q1 has orthogonal columns since it is derivedfrom Q. 
The matrix R11 is a square and upper triangular matrix. 
The diagonal entries R11 are positive since these are diagonal entries of the matrix R. 
Therefore, Q1R11 is a QR factorization of A1.

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