Science:Math Exam Resources/Courses/MATH307/December 2006/Question Section 102 07 (a)

MATH307 December 2006

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Question Section 102 07 (a)
Decide whether or not the following statement is true or false. If true, give a proof. If false, give a counterexample. (a) The product of two orthogonal matrices is 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 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
Suppose you have two orthogonal matrices and argue directly from the definition of what it means for a matrix to be orthogonal.

Hint 2
A matrix P is orthogonal if $\displaystyle PP^{T}=I$

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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. This statement is true. Proof: Let P and Q be orthogonal matrices, in particular, PTP = PPT = I, QTQ = QQT = I. Let $S=PQ$ Since the matrices P and Q are orthogonal, this means that the vectors are real. This means that $P^{*}=P^{T}$ Then ${\begin{aligned}S^{T}S&=(PQ)^{T}PQ\\&=Q^{T}P^{T}PQ\\&=Q^{T}IQ\qquad {\text{since P is orthogonal}}\\&=Q^{T}Q\\&=I\end{aligned}}$ Similarly, $SS^{T}$ = $PQ(PQ)^{T}$ = $PQQ^{T}P^{T}$ = $PIP^{T}$ = $PP^{T}$ = $I$ $\therefore S^{T}S=SS^{T}=I$ and the product of two orthogonal matrices is also orthogonal.