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Brahmagupta matrix

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

B ( x , y ) = [ x y ± t y ± x ] . {\displaystyle 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 ± t y 1 y 2 , x 1 y 2 ± y 1 x 2 ) . {\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}).\,}

Powers of the matrix are defined by

B n = [ x y t y x ] n = [ x n y n t y n x n ] ≡ B n . {\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}.}

The   x n {\displaystyle \ x_{n}} and   y n {\displaystyle \ y_{n}} are called Brahmagupta polynomials. The Brahmagupta matrices can be extended to negative integers:

B − n = [ x y t y x ] − n = [ x − n y − n t y − n x − n ] ≡ B − n . {\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}.}

See also

References

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

External links


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