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The QR Decomposition

The QR decomposition factors a matrix A into the product of an orthogonal matrix Q and an upper triangular matrix R. This decomposition can be used to solve linear systems of equations of the form Ax=B by computing Q, R, and then solving the equivalent system Ry=Q^TB. The QR algorithm is commonly used to solve linear least squares problems.

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203 views3 pages

The QR Decomposition

The QR decomposition factors a matrix A into the product of an orthogonal matrix Q and an upper triangular matrix R. This decomposition can be used to solve linear systems of equations of the form Ax=B by computing Q, R, and then solving the equivalent system Ry=Q^TB. The QR algorithm is commonly used to solve linear least squares problems.

Uploaded by

hachan
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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The QR decomposition

The QR decomposition, also known as the QR factorization, is another method of solving


linear systems of equations using matrices, very much like the LU decomposition. The
equation to solve is in the form of Ax=B, where matrix A=QR. However, in this case, A is a
product of an orthogonal matrix, Q, and upper triangular matrix, R. The QR algorithm is
commonly used to solve the linear least-squares problem.

An orthogonal matrix exhibits the following properties:

• It is a square matrix.
• Multiplying an orthogonal matrix by its transpose returns the identity matrix:

• The inverse of an orthogonal matrix equals its transpose:

An identity matrix is also a square matrix, with its main diagonal containing 1s and 0s
elsewhere.
The problem of Ax=B can now be restated as follows:

Using the same variables in the LU decomposition example, we will use theqr() method
of scipy.linalg to compute our values of Q and R, and let the y variable represent our value
of BQT with the following code:

In [ ]:
"""
QR decomposition with scipy
"""
import numpy as np
import scipy.linalg as linalg

A = np.array([
[2., 1., 1.],
[1., 3., 2.],
[1., 0., 0]])

B = np.array([4., 5., 6.])

Q, R = scipy.linalg.qr(A) # QR decomposition
y = np.dot(Q.T, B) # Let y=Q'.B

x = scipy.linalg.solve(R, y) # Solve Rx=y

Note that Q.T is simply the transpose of Q, which is also the same as the inverse of Q:

In [ ]:

print(x)

Out[ ]:

[ 6. 15. -23.]

We get the same answers as those in the LU decomposition example.

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