Section 2.1

Programming Problems

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Problem 2.101

Consider the $2 \times 2$ matrix

A(\alpha _{0}, \alpha _{1}, \alpha _{2}) = \begin{pmatrix} (\cos \alpha _{0}\cos \alpha _{1}+i\sin \alpha _{0}\sin \alpha _{1})e^{i\alpha _{2}} & -\cos \alpha _{0}\sin \alpha _{1}+i\sin \alpha _{0}\cos \alpha _{1} \\ \cos \alpha _{0}\sin \alpha _{1}+i\sin \alpha _{0}\cos \alpha _{1} & (\cos \alpha _{0}\cos \alpha _{1}-i\sin \alpha _{0}\sin \alpha _{1})e^{-i\alpha _{2}} \end{pmatrix}

where $-\pi \leq \alpha_{0} < \pi$ , $-\pi \leq \alpha_{1} < \pi$ , $0 \leq \alpha_{2} \leq \pi$ . Write a Computer algebra program in SymbolicC++ that calculates

\left. \frac{\partial }{\partial \alpha _{j}} A(\alpha _{0}, \alpha _{1}, \alpha _{2}) \right|_{\alpha _{0}=0, \alpha _{1}=0, \alpha _{2}=0}

and thus finds the generators

X_{0} = \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}, \qquad X_{1} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}, \qquad X_{2} = \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix} .

Thus

A(\alpha _{0}, \alpha _{1}, \alpha _{2}) = \exp (\alpha _{0}X_{0}+\alpha _{1}X_{1}+\alpha _{2}X_{2}) .

Problem 2.102

Let $\alpha \in \mathbb {R}$ . Given the matrix

A(\alpha ) = \begin{pmatrix} \cosh \alpha & 0 & 0 & \sinh \alpha \\ 0 & \cos \alpha & \sin \alpha & 0 \\ 0 & -\sin \alpha & \cos \alpha & 0 \\ \sinh \alpha & 0 & 0 & \cosh \alpha \end{pmatrix} .

Write a SymbolicC++ program that calculates the vector

A(\alpha ) \begin{pmatrix} x_{0} \\ x_{1} \\ x_{2} \\ x_{3} \end{pmatrix}

and shows that $x_{0}^{2}+x_{1}^{2}+x_{2}^{2}-x_{3}^{2}$ is invariant.