I know the answer is No, since you can put plus/minus on each eigenvalue. But how about putting a psd requirement? Like $A = S^2$, $S$ is psd, is $S$ unique?


I was worried about the case where if $\lambda\_1 = \lambda\_2$, then we can flip the first two columns of unitary matrix $P$. You can require all eigen values distinct. But even if has same eigen values, the matrix shall be the same?



Is the square root of a matrix unique?
A positive semidefinite matrix has a unique PSD square root - Horn&Johnson Theorem 7.2.6. Much more is know, see, for example, 


*Johnson, Charles R.; Okubo, Kazuyoshi; Reams, Robert*, [**Uniqueness of matrix square roots and an application**](http://dx.doi.org/10.1016/S0024-3795(00)00243-3), Linear Algebra Appl. 323, No.1-3, 51-60 (2001). [ZBL0976.15009](https://zbmath.org/?q=an:0976.15009). (available for free on line)


