       Re: Chebyshev's Identity

• To: mathgroup at smc.vnet.net
• Subject: [mg46521] Re: Chebyshev's Identity
• From: bobhanlon at aol.com (Bob Hanlon)
• Date: Sun, 22 Feb 2004 11:27:20 -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.