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Could someone write the proof of this thing with a reference?

mathClass
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  • this post asking about uniqueness whereas im looking for a proof for its existence – mathClass Oct 16 '18 at 12:30
  • Take an eigenbasis for the matrix, and write down how the square root matrix must act on the basis. You’ll see there is only one possible answer. – Joppy Oct 16 '18 at 12:31

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This almost follows from the answer to which @lhf was referring above: Square root of Positive Definite Matrix. Let's go:

$A$ positive definite $\Rightarrow \; \exists\, U $ unitary, $D$ diagonal with positive entries s.t. $A=UDU^{\ast}$.

Define a diagonal matrix $\sqrt{D}$ by taking positive square roots of the diagonal entries of $D$. Then, define $B:= U \sqrt{D}U^\ast$.

This $B$ will be the square root of $A$. To prove that, first compute the following: $$ B^2 = B\cdot B = ... = A\;. $$ Almost trivial hint: Use the unitary property of $U$.

It remains to show that that $B$ is as well positive definite. This follows by our construction: Could $\sqrt{D}$ have non-positive values if $A$ is positive definite? If no, this implies that $B$ is positive definite.

Marc Mingoulis
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  • $B$ is trivially PD if you know that $A$ is positive definite $\iff$ $\exists U$ unitary etc. – 5xum Oct 16 '18 at 12:52
  • Of course. Strictly speaking I would not say trivial though, since this is technically not a tautology. OP seems to be an undergrad student or not a mathematician (yet), so I tried to stress that the proof is not over once you prove $B^2=A$. These things are vital if one is on the beginning of a mathematical career. – Marc Mingoulis Oct 16 '18 at 12:58
  • Oh I totally agree with you, I just added the comment for the OP in case he had trouble with the last part. – 5xum Oct 16 '18 at 13:09