Matrix representation of the Quaternions? Can anyone explain how why the matrix representation of the quaternions using real matrices is constructed as such?
I am not sure exactly what you mean by asking "... why the ...". "Why" questions can be hard to answer satisfactory in math.

The claim is that the Quaternions $\mathbb{H}$ are isomorphic (as $\mathbb{R}$-algebras) to the given set of matrices. The isomorphism looks like this:

$$
\phi: a + bi + cj + dk \longmapsto \begin{pmatrix}a &amp; b &amp; c &amp; d  \\  -b &amp; a &amp; -d &amp; c \\ -c &amp;d &amp;a&amp; - b\\ -d&amp; -c &amp; b&amp; a\end{pmatrix}.
$$
To "understand" why this is true, you "simply" check that this is an isomorphism. 

You check for example that $\phi$ is bijective, which is clear from the construction.

Then you check that $\phi$ is an algebra homomorphism, so you need for $x,y\in \mathbb{H}$ and $\lambda \in \mathbb{R}$:


$\phi(xy) = \phi(x)\phi(y)$ for $x,y\in\mathbb{H}$
$\phi(x+y) = \phi(x) + \phi(y)$
$\phi(\lambda x) = \lambda\phi(x)$


The last two are not difficult to check. The first one requires a bit of work.

Even though this does not answer the minus signs are where there are in the matrix, I highly recommend that you try to prove that $\phi$ is a homomorphism. This exercise will make you more familiar with the Quaternions.

But note if you check property $3$ above you would need (as a special case)
$$
\phi((bi)(bi))  = \phi(ib)\phi(ib).
$$
That is you would need
$$
\begin{pmatrix} -b^2 &amp; 0 &amp; 0&amp; 0  \\  0 &amp; -b^2 &amp; 0 &amp; 0 \\ 0 &amp;0 &amp;-b^2&amp; 0\\ 0&amp; 0 &amp; 0&amp; -b^2\end{pmatrix} = \begin{pmatrix} 0 &amp; b &amp; 0&amp; 0  \\  -b &amp; 0 &amp; 0 &amp; 0 \\ 0 &amp;0 &amp;0&amp; -b\\ 0&amp; 0 &amp; b&amp; 0\end{pmatrix}\begin{pmatrix} 0 &amp; b &amp; 0&amp; 0  \\  -b &amp; 0 &amp; 0 &amp; 0 \\ 0 &amp;0 &amp;0&amp; -b\\ 0&amp; 0 &amp; b&amp; 0\end{pmatrix}.
$$
So here you can see that you "need" the minus on all the $b$'s. In this case it comes down to the fact that $i^2 = -1$.
If It is not a misunderstanding, the question of "why as such (possible others?)" is equivalent to the problem of finding out ALL 4x4 real matrix representations of the Quaternions. The fact is: all those matrix representations are conjugated to each other. In other words, this is the only 4x4 real matrix representation of the Quaternions up to equivalent.