Brahmagupta matrix

The following matrix was given by Indian mathematician Brahmagupta in 628:

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

It satisfies

B(x₁,y₁)B(x₂,y₂) = B(x₁x₂±ty₁y₂,x₁y₂±y₁,x₂). 

Powers of the matrix are defined by

$$B^n = \begin{bmatrix} x & y \\ ty & x \end{bmatrix} = \begin{bmatrix} x_n & y_n \\ ty_n & x_n \end{bmatrix} \equiv B_n.$$

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

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

See also

• Brahmagupta's identity

External links

• Eric Weisstein. Brahmagupta Matrix, MathWorld, 1999.