# Do matrices of this type form a group? [closed]

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?

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

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 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.