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 [A1A2],whereA1 has p columns. Show how to obtain a QR factorization of A1, and explain why your factorization has the appropriate properties.
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Expert Answer

Aamina Herring
Answered 2021-01-20 Author has 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:
Write the partition Q as follows:
Q=[Q1Q2],whereQ1 has p columns.
Write the partition of R as follows:
R=[R11R120R22],whereR11isap×p matrix.
Step 3
Substitute the partition values of Q and R 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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