Science:Math Exam Resources/Courses/MATH307/April 2005/Question 03 (a)

MATH307 April 2005

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[hide] Question 03 (a)
Let P be a real symmetric n x n matrix such that P2 = P. Also let R = In - 2P be the matrix that reflects a vector across a plane. (a) Show that R is an orthogonal matrix and that R2 = In.

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[show] Hint
R is an orthogonal matrix if RTR = RRT = I. Calculate these matrix products and use that P is real symmetric, i.e. PT = P.

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[show] Solution
$R^{T}R=(I-2P)^{T}(I-2P)=(I^{T}-2P^{T})(I-2P)$ Now since P is real symmetrical $P^{T}=P$ Also the transpose of the identity matrix, I is itself So expanding $(I^{T}-2P^{T})(I-2P)=I^{T}I-2PI^{T}-2P^{T}I-4P^{T}P=I^{2}+4P-4P^{2}$ As $P=P^{2}$ and $I^{2}=I$ $R^{T}R=I$ Similarily for $RR^{T}=I=(I-2P)(I-2P)^{T}=(I-2P)(I^{T}-2P^{T})$ $=II^{T}-2PI^{T}-2IP^{T}+4P^{2}I^{T}=I^{2}-2P-2P^{T}+4P^{2}=I-4P+4P=I$