Chapter 7.3, Problem 38E

Consider a rotation $T\left(\stackrel{\to }{x}\right)=A\stackrel{\to }{x}$ in ${ℝ}^3$ . (That is, A is an orthogonal $3\times 3$ matrix with determinant 1.) Show that T has a nonzero fixed point [i.e., a vector $\stackrel{\to }{\upsilon }$ with $T\left(\stackrel{\to }{\upsilon }\right)=\stackrel{\to }{\upsilon }$ ]. This result is known as Euler’s theorem, after the great Swiss mathematician Leonhard Euler (1707−1783). Hint: Consider the characteristic polynomial $f_A\left(\lambda \right)$ . Pay attention to the intercepts with both axes. Use Theorem 7.1.4.