Re: Chebyshev's Identity

*To*: mathgroup at smc.vnet.net*Subject*: [mg46515] Re: Chebyshev's Identity*From*: bobhanlon at aol.com (Bob Hanlon)*Date*: Fri, 20 Feb 2004 22:58:58 -0500 (EST)*References*: <c14spn$rna$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

By iy I assume that you meant I*y M=({{Cos[x/n]*Exp[-I*y/n],Sin[x/n]*Exp[-I*y/n]}, {-Sin[x/n]*Exp[iy/n],Cos[x/n]*Exp[I*y/n]}})^n {{(Cos[x/n]/E^((I*y)/n))^n, (Sin[x/n]/E^((I*y)/n))^n}, {((-E^(iy/n))*Sin[x/n])^n, (E^((I*y)/n)*Cos[x/n])^n}} Limit[M, n->Infinity] {{E^((-I)*y), 0}, {0, E^(I*y)}} Note that each term in the original matrix was individually raised to power n. By "matrix raised to power n" you may have meant M=MatrixPower[{{Cos[x/n]*Exp[-I*y/n], Sin[x/n]*Exp[-I*y/n]},{-Sin[x/n]*Exp[iy/n],Cos[x/n]*Exp[I*y/n]}},n]; Limit[M, n->Infinity]//FullSimplify {{Cosh[Sqrt[-x^2 - y^2]] - (I*y*Sinh[Sqrt[-x^2 - y^2]])/ Sqrt[-x^2 - y^2], (x*Sinh[Sqrt[-x^2 - y^2]])/ Sqrt[-x^2 - y^2]}, {-((x*Sinh[Sqrt[-x^2 - y^2]])/Sqrt[-x^2 - y^2]), Cosh[Sqrt[-x^2 - y^2]] + (I*y*Sinh[Sqrt[-x^2 - y^2]])/ Sqrt[-x^2 - y^2]}} Bob Hanlon In article <c14spn$rna$1 at smc.vnet.net>, "Ravinder Kumar B." <ravi at crest.ernet.in> wrote: << I have a (2x2) matrix raised to power n. M = ({{Cos[x/n]*Exp[-iy/n], Sin[x/n]*Exp[-iy/n]},{-Sin[x/n]*Exp[iy/n],Cos[x/n]*Exp[iy/n]}})^n; All I know at present is that this expression can be further simplified analytically using Chebyshev's identity to much a simpler expression in the limit n -> infinity. I am unable to find any information regarding Chebyseb's identity and its usage. Could some one please tell me more about this identity and its usage in solving above expression. Mathematica fails to do it analytically.