Brahmagupta matrix

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

[math]\displaystyle{ B(x,y) = \begin{bmatrix} x & y \\ \pm ty & \pm x \end{bmatrix}. }[/math]

It satisfies

[math]\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).\, }[/math]

Powers of the matrix are defined by

[math]\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. }[/math]

The [math]\displaystyle{ \ x_n }[/math] and [math]\displaystyle{ \ y_n }[/math] are called Brahmagupta polynomials. The Brahmagupta matrices can be extended to negative integers:

[math]\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}. }[/math]

See also

• Brahmagupta's identity
• Brahmagupta's function

External links

• Eric Weisstein. Brahmagupta Matrix, MathWorld, 1999.
• Weisstein, Eric W. (2003). CRC Concise Encyclopedia of Mathematics. Florida: CRC Press. p. 282. ISBN 1-58488-347-2.