# Iteration to find squre root of positive semidefinite matrix

Suppose matrix $A$ is positive semidefinite and $I\succeq A$. Prove that the iteration

$$Y_0=0,\hspace{3mm} Y_{n+1}=\frac{1}{2}(A+Y_n^2)$$

is nondecreasing (that is, $Y_{n+1}\succeq Y_n$ for all $n$) and converges to the matrix $I-(I-A)^{1/2}$.

My plan is to use induction, i.e., assume $Y_n\succeq Y_{n-1}$ and try to prove $Y_{n+1}\succeq Y_n$.

It is easy to see $Y_{n+1}-Y_n=\frac{1}{2}(A-I)+\frac{1}{2}(I-Y_n)^2$. It turns out to be that given $(I-Y_{n-1})^2\succeq (I-A)$, how to show $(I-Y_n)^2\succeq (I-A)$?

For the second part, I guess we need to show $\lim_{n\rightarrow\infty}(I-Y_n)^2=(I-A)$.

Can anyone give me a hint?

Hint. Orthogonally diagonalise $A$ and the problem reduces to the scalar case.