Brahmagupta Matrix

$$B(x,y)=\left[{\matrix{x & y\cr \pm ty & \pm x\cr}}\right].$$

It satisfies

$$B(x_1,y_1)B(x_2,y_2)=B(x_1x_2\pm t y_1y_2, x_1y_2\pm y_1x_2).$$

Powers of the matrix are defined by

$$B^n=\left[{\matrix{x & y\cr ty & x\cr}}\right]^n=\left[{\matrix{x_n & y_n\cr ty_n & x_n\cr}}\right]\equiv B_n.$$

The $x_n$ and $y_n$ are called Brahmagupta Polynomials. The Brahmagupta matrices can be extended to Negative Integers

$$B^{-n}=\left[{\matrix{x & y\cr ty & x\cr}}\right]^{-n}=\left... ...{x_{-n} & y_{-n}\cr ty_{-n} & x_{-n}\cr}}\right]\equiv B_{-n}.$$

See also Brahmagupta Identity

References

Suryanarayan, E. R. ``The Brahmagupta Polynomials.'' Fib. Quart. 34, 30-39, 1996.