Brahmagupta matrix

In mathematics, the following matrix was given by Indian mathematician Brahmagupta:[1]

${\displaystyle B(x,y)={\begin{bmatrix}x&y\\\pm ty&\pm x\end{bmatrix}}.}$

It satisfies

${\displaystyle B(x_{1},y_{1})B(x_{2},y_{2})=B(x_{1}x_{2}\pm ty_{1}y_{2},x_{1}y_{2}\pm y_{1}x_{2}).\,}$

Powers of the matrix are defined by

${\displaystyle B^{n}={\begin{bmatrix}x&y\\ty&x\end{bmatrix}}^{n}={\begin{bmatrix}x_{n}&y_{n}\\ty_{n}&x_{n}\end{bmatrix}}\equiv B_{n}.}$

The ${\displaystyle \ x_{n}}$ and ${\displaystyle \ y_{n}}$ are called Brahmagupta polynomials. The Brahmagupta matrices can be extended to negative integers:

${\displaystyle B^{-n}={\begin{bmatrix}x&y\\ty&x\end{bmatrix}}^{-n}={\begin{bmatrix}x_{-n}&y_{-n}\\ty_{-n}&x_{-n}\end{bmatrix}}\equiv B_{-n}.}$