Personal solution to the book Quantum Computation and Quantum Information by Michael A. Nielsen & Isaac L. Chuang, chapter 4. Currently under construction. Obviously won’t solve them all, but I’ll try my best. Open to valuable critics and better solutions. S4.2 Single qubit operations ex 4.2 method 1: use taylor expansion to check left and right are equal. method 2: Since the matrix $A$ is involutory, it is non-defective and can be eigen-decomposed with $\pm 1$ eigenvalues. $$ A = P^{-1}SP $$ in which $S$ is a signature matrix (diagonal matrix with $\pm 1$ on diagonal). Then $$ \exp(iAx)= P^{-1} \exp(iSx) P $$ For the part in the middle, $\exp(\pm ix)=\cos(x)\pm i\sin(x)$, $\exp(iSx)=\cos(x)I+i\sin(x)S$