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

MATH307 December 2008

  •  Q1  •  Q2  •  Q3  •  Q4  •  Q5  •  Q6 (a)  •  Q6 (b)  •  Q7  •  Q8  •  Q9  •  Q10  •  

   Other MATH307 Exams

  •  December 2012  •  April 2013  •  April 2012  •  April 2006  •  December 2008  •  December 2010  •  

Question 04
Find the QR decomposition of the matrix ${\begin{bmatrix}0&1\\1&0\\0&-1\\-1&0\\0&1\\\end{bmatrix}}$

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 = QTA.

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. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. 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, ${\begin{bmatrix}0\\1\\0\\-1\\0\end{bmatrix}}\cdot {\begin{bmatrix}1\\0\\-1\\0\\1\end{bmatrix}}=0.$ So we can simply put the normalized columns of A into the matrix Q: $Q={\begin{bmatrix}0&{\frac {1}{\sqrt {3}}}\\{\frac {1}{\sqrt {2}}}&0\\0&-{\frac {1}{\sqrt {3}}}\\-{\frac {1}{\sqrt {2}}}&0\\0&{\frac {1}{\sqrt {3}}}\end{bmatrix}}.$ As a last step, we calculate R = QTA to find $R={\begin{bmatrix}{\sqrt {2}}&0\\0&{\sqrt {3}}\end{bmatrix}}.$ (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.)

Click here for similar questions

MER QGH flag, MER QGQ flag, MER QGS flag, MER QGT flag, MER Tag Matrix decomposition, Pages using DynamicPageList3 parser function, Pages using DynamicPageList3 parser tag