# NCMatrix ^ ZZ -- Exponentiate an NCMatrix

## Synopsis

• Operator: ^
• Usage:
L = M^n
• Inputs:
• Outputs:

## Description

This exponentiates an NCMatrix. It should be remarked that the matrix is reduced with the GB of the ring it is over on each iteration of the product. If your algebra is significantly smaller than the tensor algebra, this is a large savings. The input is assumed to be a nonnegative integer at this time.

 i1 : A = QQ{x,y,z} o1 = A o1 : NCPolynomialRing i2 : M = ncMatrix {{x, y, z}} o2 = | x y z | o2 : NCMatrix i3 : sigma = ncMap(A,A,{y,z,x}) o3 = NCRingMap A <--- A o3 : NCRingMap i4 : N = ncMatrix {{M},{sigma M}, {sigma sigma M}} o4 = | x y z | | y z x | | z x y | o4 : NCMatrix i5 : N^3 o5 = | z^2*x+z*y*z+z*x*y+y*z*y+y^2*x+y*x*z+x*z^2+x*y^2+x^3 z^2*y+z*y*x+z*x*z+y*z^2+y^3+y*x^2+x*z*x+x*y*z+x^2*y z^3+z*y^2+z*x^2+y*z*x+y^2*z+y*x*y+x*z*y+x*y*x+x^2*z | | z^2*y+z*y*x+z*x*z+y*z^2+y^3+y*x^2+x*z*x+x*y*z+x^2*y z^3+z*y^2+z*x^2+y*z*x+y^2*z+y*x*y+x*z*y+x*y*x+x^2*z z^2*x+z*y*z+z*x*y+y*z*y+y^2*x+y*x*z+x*z^2+x*y^2+x^3 | | z^3+z*y^2+z*x^2+y*z*x+y^2*z+y*x*y+x*z*y+x*y*x+x^2*z z^2*x+z*y*z+z*x*y+y*z*y+y^2*x+y*x*z+x*z^2+x*y^2+x^3 z^2*y+z*y*x+z*x*z+y*z^2+y^3+y*x^2+x*z*x+x*y*z+x^2*y | o5 : NCMatrix i6 : B = A/ncIdeal{y*z + z*y - x^2, x*z + z*x - y^2, z^2 - x*y - y*x} --Calling Bergman for NCGB calculation. Complete! o6 = B o6 : NCQuotientRing i7 : NB = promote(N,B) o7 = | x y z | | y z x | | z x y | o7 : NCMatrix i8 : NB^3 o8 = | -y^2*z+y^3+y*x*z-y*x*y+x*y*z+x*y^2+2*x*y*x+x^2*z+3*x^2*y y^2*z+y*x*z+2*y*x*y+x*y*z+3*x*y^2-x*y*x-x^2*z+x^2*y+x^3 2*y^2*z+y^3+y*x*y+x*y*x+2*x^2*z+x^3 | | y^2*z+y*x*z+2*y*x*y+x*y*z+3*x*y^2-x*y*x-x^2*z+x^2*y+x^3 2*y^2*z+y^3+y*x*y+x*y*x+2*x^2*z+x^3 -y^2*z+y^3+y*x*z-y*x*y+x*y*z+x*y^2+2*x*y*x+x^2*z+3*x^2*y | | 2*y^2*z+y^3+y*x*y+x*y*x+2*x^2*z+x^3 -y^2*z+y^3+y*x*z-y*x*y+x*y*z+x*y^2+2*x*y*x+x^2*z+3*x^2*y y^2*z+y*x*z+2*y*x*y+x*y*z+3*x*y^2-x*y*x-x^2*z+x^2*y+x^3 | o8 : NCMatrix