Square of two positive definite matrices are equal then they are equal I have read that if $P, Q$ are two positive definite matrices such that $P^2=Q^2$, then $P=Q$.
I don't know why. Some one can help me? Thanks for any indication.

*

*If P and Q are symmetric positive definite, and $P^2=Q^2$, then the eigenvalues of the squared matrices and their eigenvectors are the same. 
(The characteristic function is the same)

*If the eigenvalues of $P^2$ are all positive, the eigenvalues of $P$ are merely the (real) squareroots of the eigenvalues of $P^2$.

*The eigenvectors of P and Q are the same as that of $P^2$ and $Q^2$. 

*By constructing P and Q using the $SAS^{-1}$ form, you can prove that they are equal.

*Note that symmetric matrices are never defective.

