Past exam of the mathematics course of the University of Cambridge / 2019 / ia / Paper 1 / 8A (Vectors and Matrices)

LetSince $J_{-}^2=-I$ and $J_{+}^2=I$, separating the even and odd terms of the matrix exponential givesand

Direct multiplication and the identities ${\cos }^2\theta +{\sin }^2\theta =1$ and ${\cosh }^2\theta -{\sinh }^2\theta =1$ giveThus $R$ is an orthogonal matrix, while $S$ preserves the indefinite quadratic form $x_1^2-x_2^2$.

The matrix $A$ fixes the first coordinate and restricts to one half of the rotation generator on the last two coordinates. Hence

The matrix $B$ fixes the second coordinate and satisfies the hyperbolic-generator relation on the first and third coordinates. Therefore

The rotation matrix $e^{xA}$ commutes with $C$ and is orthogonal, soSince $C^n=\mathrm{diag}(1,(-1{)}^n,(-1{)}^n)$,

On the first and third coordinates, the identity $SJS=J$ from part (b) gives $e^{xB}C{e}^{xB}=C$; the second coordinate is also unchanged. Thus