## How is the algebra of quaternions isomorphic to the algebra of matrices?

I just started learning about morphisms and I came across a problem that totally stumps me. Here goes:

Show that the algebra of quaternions is isomorphic to the algebra of matrices of the form:

\begin{pmatrix}
\alpha & \beta \\
-\bar{\beta} & \bar{\alpha} \end{pmatrix}

where α,β$\in$ℂ and the overbar indicates complex conjugation.

[Hint: If q=a+bi+cj+dk is a quaternion, identify it with the matrix whose entries are $\alpha$ =a+bi, $\beta$ =c+di, $\bar{\alpha}$ =a-bi, -$\bar{\beta}$ =-c+di, (a,b,c,d) $\in$ℝ, i$^{2}$=j$^{2}$=k$^{2}$=-1]

Thanks a lot everyone!
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 Quote by mitch_jacky I just started learning about morphisms and I came across a problem that totally stumps me. Here goes: Show that the algebra of quaternions is isomorphic to the algebra of matrices of the form: \begin{pmatrix} \alpha & \beta \\ -\bar{\beta} & \bar{\alpha} \end{pmatrix} where α,β$\in$ℂ and the overbar indicates complex conjugation. [Hint: If q=a+bi+cj+dk is a quaternion, identify it with the matrix whose entries are $\alpha$ =a+bi, $\beta$ =c+di, $\bar{\alpha}$ =a-bi, -$\bar{\beta}$ =-c+di, (a,b,c,d) $\in$ℝ, i$^{2}$=j$^{2}$=k$^{2}$=-1] Thanks a lot everyone!

What have you tried? Have you already summed (multiplied) two of the above matrices and verified the sum behaves as the

sum (product) of the respective quaternions?

For example, heeding to the hint given, we'd have the identifications

$$i \sim \begin{pmatrix}1&0\\0&-1\end{pmatrix}\,\,,\,j \sim \begin{pmatrix}0&1\\-1&0\end{pmatrix}\,\,,\,k \sim \begin{pmatrix}0&1\\1&0\end{pmatrix}$$so do we have

$$ij=k \sim \begin{pmatrix}1&0\\0&-1\end{pmatrix} \begin{pmatrix}0&1\\-1&0\end{pmatrix}\stackrel{?}=\begin{pmatrix}0&1\\1&0\end{pmatrix}\,\, ??$$

DonAntonio
 Okay, I have verified that ij=k but I don't see what to do next.

## How is the algebra of quaternions isomorphic to the algebra of matrices?

The more I look at this problem the less I understand what I am supposed to be doing. Without giving me any answers what I need is for someone to walk me through the steps to the solution. I need an understanding of what I'm trying to solve here.

Recognitions:
Homework Help
 Quote by mitch_jacky I just started learning about morphisms and I came across a problem that totally stumps me. Here goes: Show that the algebra of quaternions is isomorphic to the algebra of matrices of the form: \begin{pmatrix} \alpha & \beta \\ -\bar{\beta} & \bar{\alpha} \end{pmatrix} where α,β$\in$ℂ and the overbar indicates complex conjugation. [Hint: If q=a+bi+cj+dk is a quaternion, identify it with the matrix whose entries are $\alpha$ =a+bi, $\beta$ =c+di, $\bar{\alpha}$ =a-bi, -$\bar{\beta}$ =-c+di, (a,b,c,d) $\in$ℝ, i$^{2}$=j$^{2}$=k$^{2}$=-1] Thanks a lot everyone!
What does it MEAN for two algebraic structures to be isomorphic? (That is, there is a formal definition of isomorphism, and I am asking you what that is.) Then you need to show that all the parts of the definition hold true.

RGV
 Okay, so an isomorphism is a bijectivity from one structure to another which preserves the characteristics of the structures. So if quaternions possess addition, scalar and quaternion multiplication I must show that this structure is isomorphic to the addition, scalar, and matrix multiplication of matrix algebra?

Recognitions:
Homework Help
 Quote by mitch_jacky Okay, so an isomorphism is a bijectivity from one structure to another which preserves the characteristics of the structures. So if quaternions possess addition, scalar and quaternion multiplication I must show that this structure is isomorphic to the addition, scalar, and matrix multiplication of matrix algebra?
Well, what do YOU think?

RGV
 You know, I think that's pretty much what I am being asked to show. I appreciate your insights! So basically (and please excuse the poor notation, I'm on my phone): (H,\otimes,°,\oplus, =)~(M, ×, •, +, =)

Recognitions:
Homework Help
 Quote by mitch_jacky You know, I think that's pretty much what I am being asked to show. I appreciate your insights! So basically (and please excuse the poor notation, I'm on my phone): (H,\otimes,°,\oplus, =)~(M, ×, •, +, =)
I guess so, depending on exactly what the symbols mean; anyway, if q1 ⇔ A1 and q2 ⇔ A2, you need q1+q2 ⇔ A1+A2, q1*q2 ⇔ A1.A2 (matrix product) and, for scalar c, cq1 ⇔ cA1, etc.

RGV
 Wow, it isn't wizard math after all. Thanks man!

 Tags isomorphism, quaternion