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For any real matrix $A$, write down a sufficient condition for a real matrix $E$ to exist such that $E^2 = A$, and prove that this condition is sufficient.

How would I go about answering this question? I need a starting point please

Dietrich Burde
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Ellie
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  • What is E2? Is E the identity matrix? – Mike Jan 15 '20 at 19:40
  • You need to find sufficients conditions for a real matrix $E$ so that $E^2 = A$? – azif00 Jan 15 '20 at 19:43
  • I’m not given anything for E either so I’m not really sure what’s going on – Ellie Jan 15 '20 at 19:46
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    When you are given an exercise so badly thought take full advantage of whoever wrote it and answer it to your convenience. They only asked for sufficients conditions. So, you could give a condition as restrictive as you want. For example, $A=0$ is a sufficient condition. If satisfied, there is $E$, the matrix $E=0$, such that $E^2=A$. – MoonLightSyzygy Jan 15 '20 at 19:52
  • Many answers have been given here already. Have a look, e.g., starting here. – Dietrich Burde Jan 15 '20 at 20:35
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    @DietrichBurde Note that the OP wants a square root over $\mathbb R$, not $\mathbb C$. – Robert Israel Jan 15 '20 at 21:04

1 Answers1

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Hint: you might start by investigating the case of diagonal matrices. Then generalize a bit to diagonalizable ones.

On the other hand, examples such as $\pmatrix{-1 & 0\cr 0 & 1\cr}$ and $\pmatrix{0 & 1\cr 0 & 0\cr}$ that have no real square root show you can't generalize too far.

Robert Israel
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