*The main question here is:*

*
**given a matrix then what is ?*

Given is the matrix

(1) . . . .

It is clear that

This can be written as . Note that and . We shall write and . Then we obtain

(2) . . . .

The matrix has eigen-values , defined by . This gives the equation

(3) . . . .

Then we obtain , so

(4) . . . .

It is clear that

(5a) . . .

and

(5b) . . . .

Therefore we can write equation (2) as

(6) . . .

We now define

(7) . . . .

It is clear that

,

therefore

,

so we can write

(8) . . .

It is also clear that

and

.

Therefore we obtain

(9a) . . .

and

(9b) . . . .

As it is clear that

(10) . . .

It is als clear that IF

and

then

and as we obtain

.

As and we obtain

So the general result is

(11) . . .

From this follows

Given any 2 × 2 matrix

, where

,

then

,

where

,

where

and

.

We have found that , therefore we can write . Then we find that and . We obtain

(12) . . . .

As we obtain , therefore we can write and . Then we obtain , therefore we can write and . Then

And when we put this in equation (12) we get

(13) . . .

as the final result.

It is also clear that the one-parameter group of 2 × 2 matrices takes the form

where is the group-parameter.

Hope you liked this post!

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very nice theorem … i m working on representation theory of some groups in SL2(Z) and this is a wonderful tool ! thanks !

Mohamed,

You are welcome, I am working on a general theorem for nxn matrix…

I worked out explicit formulae for n x n matrix powers since 1989. A PDF copy of my typewritten article is available. aldo.boiti@iol.it

I liked very much your post “Raising the power of a 2 x 2 matrix”, of December 3, 2011. Very

interesting exposition.

In the little known mathematical journal “L’Insegnamento della Matematica e delle Scienze integrate”,

Vol. 12, n. 2, February 1989, p. 238-243, in the article “Boiti A. Le potenze di una matrice quadrata ed

il teorema del punto unito: applicazioni”, besides the explicit formulae for the nth powers of 2 x 2

matrices with both different and coinciding eigenvalues you also find the explicit formulae for the nth

powers of 3 x 3 matrices, with different eigenvalues. The link is

“http://www.centromorin.it/home/pubblicazioni/riviste/tabanni.asp”. Select 1989 and 2 or Febbraio. As

of June 2015 the password to access the article is “Chuquet1488”. A better copy of the article, in PDF

format, may be obtained from “aldo.boiti@iol.it”.

3 x 3 matrix B = (bi,j), with three different eigenvalues vi; i, j = 1, 2, 3.

B^n = enB^2 + fnB + gnI ; I = unit matrix.

en = c123 + c231 + c312; cijk = vi^n /((vi – vj)(vi – vk)).

fn = (d123 + d231 + d312)/q.

dijk = vi^n (vj^2 – vk^2).

q = (v1 – v2)(v2 – v3)(v3 – v1).

gn = v1v2v3 en-1. (my formulae since 1989)