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Unformatted text preview: e of conic sections, the orthogonal matrix B of determinant
one which relates x and y represents a rotation. To see this, note ﬁrst that since
B is orthogonal,
(B x) · (B y) = xT B T B y = xT I y = x · y. (2.8) In other words, multiplication by B preserves dot products. It follows from this
that the only real eigenvalues of B can be ±1. Indeed, if x is an eigenvector for
B corresponding to the eigenvalue λ, then
λ2 (x · x) = (λx) · (λx) = (B x) · (B x) = x · x,
40 so division by x · x yields λ2 = 1.
Since det B = 1 and det B is the product of the eigenvalues, if all of the
eigenvalues are real, λ = 1 must occur as an eigenvalue. On the other hand,
non-real eigenvalues must occur in complex conjugate pairs, so if there is a nonreal eigenvalue µ + iν , then there must be another non-real eigenvalue µ − iν
together with a real eigenvalue λ. In this case,
1 = det B = λ(µ + iν )(µ − iν ) = λ(µ2 + ν 2 ).
Since λ = ±1, we conclude that λ = 1 must occur as an...
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This document was uploaded on 01/12/2014.
- Winter '14