Science:Math Exam Resources/Courses/MATH307/December 2008/Question 04
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Question 04 

Find the QR decomposition of the matrix 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

What do you notice when you take the dot product of the two columns of A? How does this simplify the computation? 
Hint 2 

Since the two columns of A are already orthogonal, we can simply use the normalized columns of A in Q. 
Hint 3 

As a last step, since A = QR and Q has orthonormal columns, we can find R by computing R = Q^{T}A. 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. To begin with, we note that the columns of A are already orthogonal, So we can simply put the normalized columns of A into the matrix Q: As a last step, we calculate R = Q^{T}A to find (Note that the question asks for the QR decomposition, not a QR decomposition. This is the unique QR decomposition of A such that Q has orthonormal columns and R is square upper triangular with positive diagonal entries.) 