Chapter 5.3, Problem 64E

This exercise shows one way to define the quaternions, discovered in 1843 by the Irish mathematician Sir W. R. Hamilton (1805-1865). Consider the set H of all $4\times 4$ matrices M of the form
   $M=\left[\begin{array}{cccc}p& -q& -r& -s\\ q& p& s& -r\\ r& -s& p& q\\ s& r& -q& p\end{array}\right]$ ,
where p, q, r, s are arbitrary real numbers. We can write M more succinctly in partitioned form as
   $M=\left[\begin{array}{cc}A& -B^T\\ B& A^T\end{array}\right]$ ,
where A and B arc rotation-scaling matrices.

a. Show that H is closed under addition: If M and N arc in H, then so is $M+N$ .<
b. Show that H is closed under scalar multiplication: If M is in H and k is an arbitrary scalar, then kM is in H.
c. Parts (a) and (b) show that H is a subspace of the linear space ${ℝ}^{4\times 4}$ . Find a basis of H, and thus determine the dimension of H.
d. Show that H is closed under multiplication: If M and N are in H, then so is MN.
e. Show that if M is in H, then so is $M^T$ .
f. For a matrix M in H. compute $M^{T}M$ .
g. Which matrices M in H are invertible? If a matrix M in H is invertible, is $M^{-1}$ necessarily in H as well?
h. If M and N are in H, does the equation $MN=NM$ always hold?