Brahmagupta matrix

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


 * $$B(x,y) = \begin{bmatrix}

x & y \\ \pm ty & \pm x \end{bmatrix}.$$

It satisfies


 * $$B(x_1,y_1) B(x_2,y_2) = B(x_1 x_2 \pm ty_1 y_2,x_1 y_2 \pm y_1 x_2).\,$$

Powers of the matrix are defined by


 * $$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 $$\ x_n$$ and $$\ y_n$$ are called Brahmagupta polynomials. The Brahmagupta matrices can be extended to negative integers:


 * $$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}.$$