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Powers of matrices equal to the identity matrix

by Bipolarity
Tags: equal, identity, matrices, matrix, powers
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Bipolarity
#1
Nov18-12, 02:36 PM
P: 783
I am curious about under what conditions the powers of a square matrix can equal the identity matrix.

Suppose that A is a square matrix so that [itex] A^{2} = I [/itex]

At first I conjectured that A is also an identity matrix, but I found a counterexample to this.
I noticed that the counterexample was an elementary matrix.

So then I conjectured that A is an elementary matrix. Is this true? Can I prove this? What about for general powers of A?

BiP
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AlephZero
#2
Nov18-12, 03:54 PM
Engineering
Sci Advisor
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Thanks
P: 7,147
As a simple example think about 2x2 matrices.

If ##\displaystyle A = \begin{bmatrix}a & b \\ c & d \end{bmatrix}##, then

##\displaystyle A^2 = \begin{bmatrix}a^2 + bc & b(a+d) \\ c(a+d) & d^2 + bc \end{bmatrix} = I##.

From the off-diagonal terms, ##b(a+d) = 0## and ##c(a+d) = 0##.
Taking ##b = c = 0## isn't going to lead to anywhere interesting, so let's see what happens if ##d = -a##.
From the diagonal terms, ##a^2 + bc = 1##.

You can satisfy that with matrices that are not elementary, for example
##\displaystyle A = \begin{bmatrix} 2 & 3 \\ -1 & -2 \end{bmatrix}##.

In fact the condition ##a^2 + bc = 1## here is the same as ##|\det A| = 1##, which isn't a complete coincidence - but things are not so simple for bigger matrices.
Bipolarity
#3
Nov18-12, 09:06 PM
P: 783
I see. Thanks much [itex]\aleph_0[/itex]

BiP

Norwegian
#4
Nov19-12, 01:26 AM
P: 144
Powers of matrices equal to the identity matrix

A solution to An=I is obviously attained if A is a suitable diagonal or rotation matrix, and also for all similar matrices PAP-1, where P is invertible.


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