Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top
Maybe helpful: Think of the Quaternions as a four dimensional vector space over $\mathbb{R}$ with basis ${1,i,j,k}$. Then think of the natural action of the Quatenions on this vector space. The matrices you get for the actions of $1,i,j,k$ are just the ones you found in Wikipedia. – user3533 Jan 26 '13 at 21:30
Could you explain what a natural action is? I'm unfamiliar with the term. – chubbycantorset Jan 26 '13 at 21:35
The action of multiplication. Let's take the case of multiplication by $i$ on the left ($x\mapsto ix$). This is a linear action on this vector space, so it can be represented by a matrix relative to the basis ${1,i,j,k}$. Find this matrix. – user3533 Jan 26 '13 at 21:39
@user3533: How do you convert a linear action into a matrix? – chubbycantorset Jan 26 '13 at 21:47
The usual way you represent a linear operator as a matrix. In the first column put the coefficients of $i \cdot 1 = i$, in the second column, those of $i \cdot i = -1$, in the third those of $i \cdot j = k$ and in the forth those of $i \cdot k = -j$ – user3533 Jan 26 '13 at 21:56

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:

$$ \phi: a + bi + cj + dk \longmapsto \begin{pmatrix}a & b & c & d \\ -b & a & -d & c \\ -c &d &a& - b\\ -d& -c & b& a\end{pmatrix}. $$ 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 $\lambda \in \mathbb{R}$:

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

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 $$ \begin{pmatrix} -b^2 & 0 & 0& 0 \\ 0 & -b^2 & 0 & 0 \\ 0 &0 &-b^2& 0\\ 0& 0 & 0& -b^2\end{pmatrix} = \begin{pmatrix} 0 & b & 0& 0 \\ -b & 0 & 0 & 0 \\ 0 &0 &0& -b\\ 0& 0 & b& 0\end{pmatrix}\begin{pmatrix} 0 & b & 0& 0 \\ -b & 0 & 0 & 0 \\ 0 &0 &0& -b\\ 0& 0 & b& 0\end{pmatrix}. $$ 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 $i^2 = -1$.

share|cite|improve this answer
The question asks about the real matrix representation; not the complex matrix representation. And by "why", I mean to say that surely that 4x4 matrix wasn't constructed accidentally. What is the logic behind its construction? Why are the minus signs and letters $a,b,c,d$ placed where they are? – chubbycantorset Jan 26 '13 at 21:21
@chubbycantorset: I updated my answer. Does it help now? – Thomas Jan 26 '13 at 21:26
Not exactly. I've checked those things before, but my question asks how the map was constructed as such. To make an analogy, perhaps the formula to add the first $n$ digits was created by "experimental observation," so to say. Maybe someone just looked at the sequence of sums, and found a pattern to deduce that it should be $n(n+1)/2$. I'm looking for a similar explanation here, since I can't see how you can "experimentally observe" that this has to be the matrix for the quaternions. – chubbycantorset Jan 26 '13 at 21:33
@chubbycantorset: So you are asking for how people historically came up with the representation? – Thomas Jan 26 '13 at 21:36
How they may have come up with it. Not necessarily how they did it exactly. I want to know if observing some property of the Quarternions can lead one to deduce that this should be the representation matrix. Although the placement of the minus signs on $b$ does help somewhat understand what's going on. – chubbycantorset Jan 26 '13 at 21:41

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.