What is the proof of the fact "a positive definite matrix has a unique positive definite square root"?

Question

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

Answer 1

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.