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It's too easy question but I'm confused.

Let A matrix is \begin{pmatrix} 5 & 4\\ 4 & 5\\ \end{pmatrix}

Of course it can be diagonalized since it has distinct two eigenvalues.

The question is to find square roots, that is $R^2$=A.

I think there are two possible $R$, $A^\frac{1}{2}$ and $-A^\frac{1}{2}$

However the answer is four. What are the other two?

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1 Answer 1

up vote 1 down vote accepted

Hint: $A$ has two distinct eigenvalues, and both of them have two square roots.

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Two eigenvalues are 1 and 9, so their square roots are 1 and 3. But the combinations are only two cases, (1,3) (-1,-3). Is is possible to combine (-1,3) (1,-3)? I tried but they can't make it. –  email Nov 11 '12 at 13:31
    
@email: Yes, there are four combinations as you said. What is your problem? –  23rd Nov 11 '12 at 13:37
    
Oh, I made some calculation error. Now I find four combinations. Thanks! –  email Nov 11 '12 at 13:48

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