$A = B^2$ for which matrix $A$? Is it true that for any $A\in M(n,\mathbb{C})$there exist a $B\in M(n,\mathbb{C})$ such that $A = B^2$? I think this is not true (but I don't know nay example), and then is it possible to characterize such $A$? 

The Cholesky decomposition is loosely related to the concept of taking the square root of a matrix. If $A$ is a positive-definite Hermitian matrix, then $$ A = B B^{*} $$, where $B$ is lower triangular with positive diangonal elements where $B^*$ is the conjugate transpose of B.
