# Math Help - Orthogonal matrix

1. ## Orthogonal matrix

I need to prove/disprove that if A^2 is orthogonal matrix, than A is also orthogonal.

I think the statement is true, because if A^2 orthogonal, then A^tA^t = A^-1A^-1, hence AAA^tA^t = AAA^-1A^-1 = AIA^-1 = AA^-1 = I, hence A is orthogonal.

Is my proof right?

2. ## Re: Orthogonal matrix

Originally Posted by rebecca
I need to prove/disprove that if A^2 is orthogonal matrix, than A is also orthogonal.

I think the statement is true, because if A^2 orthogonal, then A^tA^t = A^-1A^-1, hence AAA^tA^t = AAA^-1A^-1 = AIA^-1 = AA^-1 = I, hence A is orthogonal.

Is my proof right?
Your first statement doesn't necessarily follow.

We know that

$(AA)^T(AA)=(AA)(AA)^T=I$

and

$(AA)^T=A^TA^T$

so

$(AA)^{-1}=(A^TA^T)$

but we don't know that $A^T=A^{-1}$ yet

3. ## Re: Orthogonal matrix

You're going to have a hard time proving this, because it's not true.

Consider the matrix:

$A = \begin{bmatrix}1&2\\-1&-1\end{bmatrix}$

Now $A^2 = -I$ and $-I$ is an orthogonal matrix:

$(-I)^T = -I$ and $(-I)(-I)^T = (-I)(-I) = I$

However, $A$ is not orthogonal.

(aa)(aa)'=i
(aa)(a'a')=i
a[(aa')a']=i
(aa')a'=a'
aa'=i

5. ## Re: Orthogonal matrix

Originally Posted by Hartlw
(aa)(aa)'=i
(aa)(a'a')=i
a[(aa')a']=i

(aa')a'=a'

how do you justify this step?

a'a != I

6. ## Re: Orthogonal matrix

Originally Posted by romsek
[/B]

how do you justify this step?

a'a != I
I don't.
(aa')a'=a-1
You're right. You don't know a'=a-1

7. ## Re: Orthogonal matrix

(AB)-1=B-1A-1
A'A'=A-1A-1
if A2=B2 CAN YOU SHOW A=+-B ? I can't.

8. ## Re: Orthogonal matrix

Originally Posted by Hartlw
(AB)-1=B-1A-1
A'A'=A-1A-1
if A2=B2 CAN YOU SHOW A=+-B ? I can't.
Deveno already gave a counter example. The original assertion isn't true.

9. ## Re: Orthogonal matrix

Originally Posted by romsek
Deveno already gave a counter example. The original assertion isn't true.

if A2=B2 CAN YOU SHOW A=+-B ? If you can, A'=+-A-1, which is an interesting result. Guess I haven't struck an intellectual curiousity chord.
I'll have to get the answer myself.

10. ## Re: Orthogonal matrix

Originally Posted by Hartlw

if A2=B2 CAN YOU SHOW A=+-B ? If you can, A'=+-A-1, which is an interesting result. Guess I haven't struck an intellectual curiousity chord.
I'll have to get the answer myself.
I admit I didn't read closely enough to see that you were asking a general question. I will take a look.

11. ## Re: Orthogonal matrix

Does X2=A2 → X=±A?
No, because matrices have divisors of 0. Explanation:
X2-A2=(X-A)(X+A)=0 does not imply X-A=0 or X+A=0 for matrices.

If you prefer a counter example, that allegedly has been done for a 2X2 matrix (I tend not to work through Deveno’s posts, but I’ll take romseks’ word for it).

OK, I’m happy. My question is answered. Since everyone else was apparently happy before my posts, we are all happy now.

EDIT: In answer to the OP, one could say if A2 is orthogonal, A could be orthogonal, but not necessarily.