8. it is proved that each semi-positive definite matrix has a unique square root of the semi-positive definite number. That is, if $ A \ geq 0 $, there is a unique $ B \ geq 0 $ that satisfies $ B ^ 2 = A $.
Proof: $ A \ geq 0 $ indicates the existence of U $, so that $ \ Bex U ^ * Au = \ diag (\ lm_1, \ cdots, \ lm_n ), \ quad \ lm_ I \ geq 0. \ EEx $ \ Bex B = U \ diag (\ SQRT {\ lm_1}, \ cdots, \ SQRT {\ lm_n}) U ^ *, \ EEx $ B \ geq 0 $, and $ A = B ^ 2 $. forward certificate uniqueness. if $ C \ geq 0 $ is also suitable for $ A = C ^ 2 $, then $ \ Bex c = V \ diag (\ SQRT {\ lm_1}, \ cdots, \ SQRT {\ lm_n}) V ^ *. \ EEx $ \ Bex U \ diag (\ lm_1, \ cdots, \ lm_n) U ^ * = B ^ 2 = A = C ^ 2 = V \ diag (\ lm_1, \ cdots, \ lm_n) V ^ *, \ EEx $ \ Bex w \ diag (\ lm_1, \ cdots, \ lm_n) = \ diag (\ lm_1, \ cdots, \ lm_n) W, \ quad W = V ^ * U. \ EEx $ $ \ Bex W = \ diag (W_1, \ cdots, w_s) \ EEx $ quasi-diagonal arrays, $ w_ I $ corresponds to the same $ \ lm_j $. in this case, $ \ bee \ label {3_8_sqrt} w \ diag (\ SQRT {\ lm_1}, \ cdots, \ SQRT {\ lm_n }) = \ diag (\ SQRT {\ lm_1}, \ cdots, \ SQRT {\ lm_n}) W, \ EEE $ \ beex \ Bea B & = U \ diag (\ SQRT {\ lm_1}, \ cdots, \ SQRT {\ lm_n }) U ^ * \ & = VW \ diag (\ SQRT {\ lm_1}, \ cdots, \ SQRT {\ lm_n }) w ^ * V ^ * \ & = V \ diag (\ SQRT {\ lm_1}, \ cdots, \ SQRT {\ lm_n }) V ^ * \ quad \ sex {\ eqref {3_8_sqrt }}\\& = C. \ EEA \ eeex $
[Zhan Xiang matrix theory exercise reference] exercise 3.8