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

MATH307 December 2008

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Question 06 (a)
Let u be a unit vector and Q = I - 2uuT. 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 QT = Q. Remember that (ab)T = bTaT. To show that Q is orthogonal, confirm that QTQ = I. Use that uTu = ||u||2 and that u is a unit vector.

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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. First, we can show that the matrix Q is symmetric by showing that Q=QT: $\displaystyle Q^{T}=(I-2uu^{T})^{T}=I^{T}-(2uu^{T})^{T}=I-2(u^{T})^{T}u^{T}=I-2uu^{T}=Q$ Next, we can show the matrix Q is orthogonal by showing that QTQ=I. Since we know that Q is symmetric, QTQ=QQ. ${\begin{aligned}Q^{T}Q&=(I-2uu^{T})(I-2uu^{T})=I-4uu^{T}+4(uu^{T})(uu^{T})\\&=I-4uu^{T}+4u(u^{T}u)u^{T}\end{aligned}}$ Notice that uTu is just a number. In fact, $u^{T}u=\|u\|^{2}$ and since $u$ is unit vector $\|u\|^{2}=1$. Thus, indeed Q is orthonogal: $\displaystyle Q^{T}Q=I-4uu^{T}+4u(1)u^{T}=I-4uu^{T}+4uu^{T}=I.$

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