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

  • $\begingroup$ What is E2? Is E the identity matrix? $\endgroup$
    – Mike
    Jan 15 '20 at 19:40
  • $\begingroup$ You need to find sufficients conditions for a real matrix $E$ so that $E^2 = A$? $\endgroup$
    – azif00
    Jan 15 '20 at 19:43
  • $\begingroup$ I’m not given anything for E either so I’m not really sure what’s going on $\endgroup$
    – Ellie
    Jan 15 '20 at 19:46
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    $\begingroup$ 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 $E^2=A$. $\endgroup$ Jan 15 '20 at 19:52
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    $\begingroup$ @DietrichBurde Note that the OP wants a square root over $\mathbb R$, not $\mathbb C$. $\endgroup$ Jan 15 '20 at 21:04

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.

  • $\begingroup$ So I would need a diagonal matrix that i can take the square route of? $\endgroup$
    – Ellie
    Jan 15 '20 at 20:36
  • $\begingroup$ That's a start, yes. $\endgroup$ Jan 15 '20 at 21:05
  • $\begingroup$ Thank you! Would that be my A or would I call that D. I know $C^-1AC$ $\endgroup$
    – Ellie
    Jan 15 '20 at 21:06
  • $\begingroup$ If you call the diagonal matrix $D$, under what condition does $D$ have a (real) square root? If $D$ has a square root and $D = C^{-1} A C$, does $A$ have a square root? $\endgroup$ Jan 15 '20 at 21:42
  • $\begingroup$ I honestly don’t know😂 my heads fried $\endgroup$
    – Ellie
    Jan 15 '20 at 21:43

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