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

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[itex]\in[/itex]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 [itex]\alpha[/itex] =a+bi, [itex]\beta[/itex] =c+di, [itex]\bar{\alpha}[/itex] =a-bi, -[itex]\bar{\beta}[/itex] =-c+di, (a,b,c,d) [itex]\in[/itex]ℝ, i[itex]^{2}[/itex]=j[itex]^{2}[/itex]=k[itex]^{2}[/itex]=-1]

Thanks a lot everyone!

DonAntonio

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

[tex]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}[/tex]so do we have

[tex]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}\,\, ??[/tex]

DonAntonio

mitch_jacky

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

mitch_jacky

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.

Ray Vickson

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

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?

Ray Vickson

Well, what do YOU think?

RGV

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, ×, •, +, =)

Ray Vickson

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

mitch_jacky

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