Introduction to Computational Linear Algebra

94 Introduction to Computational Linear Algebra

Theorem 4.9 Let α, β ∈ R such that β 6= 0. There exist two rotations

R =



c −s

s c



with c = cos θ and s = sin θ such that R



α

β



=



±ρ

0



with ρ =

p

α

2

+ β

2

. One of them satisfies θ ∈ [−

π

2

,

π

2

). It is determined by:

"

if |β| > |α|, τ = −

α

β

, s =

1

√

1+τ

2

, c = sτ,

else τ = −

β

α

, c =

1

√

1+τ

2

, s = cτ.

(4.38)

Proof. A rotation is a solution of the problem if and only if sα + cβ = 0. This

condition is equivalent to cot θ = −

α

β

. As soon as such a rotation is chosen,

the first relation cα −sβ = ±ρ is automatically satisfied since a rotation is an

isometry and therefore kR



α

β



k = ρ. There are two possible values for θ

in the interv