Science:Math Exam Resources/Courses/MATH307/December 2008/Question 06 (a)
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Question 06 (a) 

Let u be a unit vector and Q = I  2uu^{T}. Show: (a) The matrix Q is symmetric and orthogonal. 
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 hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

To show that Q is symmetric, confirm that Q^{T} = Q. Remember that (ab)^{T} = b^{T}a^{T}. To show that Q is orthogonal, confirm that Q^{T}Q = I. Use that u^{T}u = u^{2} and that u is a unit vector. 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. First, we can show that the matrix Q is symmetric by showing that Q=Q^{T}: Next, we can show the matrix Q is orthogonal by showing that Q^{T}Q=I. Since we know that Q is symmetric, Q^{T}Q=QQ. Notice that u^{T}u is just a number. In fact, and since is unit vector . Thus, indeed Q is orthonogal: 