Brahmagupta matrix

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

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

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:

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

See also

Brahmagupta's identity
Brahmagupta's function

References

"The Brahmagupta polynomials" (PDF). Suryanarayanan. The Fibonacci Quarterly. Retrieved 3 November 2011.

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.

Mathematics Encyclopedia