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I have a set of matrices of the form $$A = \begin{pmatrix} a & b\\ -b & a \end{pmatrix}$$ where $a,b \in \mathbb{C}$. They also have the property $$A A^{T} = (a^{2} + b^{2})\mathbb{I}$$ We do know that they form a group with two conditions:

  • $a^{2} + b^{2} \neq 0$
  • ${|a| }^{2} + {|b|}^{2} \neq 0$

What group do these matrices form?

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    $\begingroup$ If $a,b \in \mathbb{R}$, then the matrix $A$ represents the complex number $a+bi$. $\endgroup$
    – Geoffrey Trang
    Jul 14, 2021 at 18:11
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    $\begingroup$ why are there two reopen votes? $\endgroup$
    – Calvin Khor
    Jul 15, 2021 at 1:14

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The Wikipedia article Bicomplex number has the key to the answer. It states that

The general bicomplex number can be represented by the matrix ${w\: iz \choose iz \:w}$ which has determinant $w^2+z^2$.

Use a different basis and the matrix becomes ${a\:\: b \choose -b \:a}$ which has determinant $a^2+b^2$ which form a complex algebra which is a ring because they are closed under addition and multiplication with basis $\,\{I,J\}\,$ where $$I^2=I,\,IJ=JI=J,\,J^2=-I.$$ One matrix representation is $$ I={1\:0 \choose 0\:1},\quad J={0\:1 \choose -1\:0} $$ and another is $$ I={1\:0 \choose 0\:1},\quad J={0\:i \choose i\:0} . $$

The Wikipedia article also states

The bicomplex numbers form a commutative algebra over $\,\mathbb{C}\,$ of dimension two, which is isomorphic to the direct sum of algebras $\,\mathbb{C}\oplus\mathbb{C}.$

The isomorphism is similar to the case of split-complex numbers $$aI+bJ\;\;\leftrightarrow\;\;(a+b\,i,a-b\,i)$$ where addition and multiplication are defined componentwise which implies that the group of units is $\,\mathbb C^{\times}\times\mathbb C^{\times}$ with all numbers of the form $\,a\,I\pm a\,i\,J\,$ being the zero divisors.

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