I'm working on a problem and was lead to trying to find an approximation for the square root of a matrix. I came across a way of doing this using [holomorphic functional calculus][1]. However, my first attempt at a computation resulted in failure. Did I miss something? --

Wikipedia gives the expansion of the square root matrix as follows:
$$\frac{A^{1/2}}{\|A\|^{1/2}} = I - \sum_{n = 1}^{\infty}\left\lvert {1/2 \choose n}\right\rvert  \left(I-\frac{A}{\|A\|}\right)^n = I - \frac 1 2\left(I-\frac{A}{\|A\|}\right) - \frac 1 8 \left(I-\frac{A}{\|A\|}\right)^2 \,...$$
In this post, I'm taking the matrix norm to be the max of the absolute values of the column-sums of matrix $A$.


I thought I might be able to get an estimate of square root matrix by taking the first two terms and using the third term as an error bound. But a simple numerical example appears to not work. I'm not sure if I'm using the expansion incorrectly, or there's some hypothesis I'm missing. In particular for $A = \begin{pmatrix}
4 & 0 \\
0 & 16 
\end{pmatrix}, \|A\| = 16$.

I get an error:$$\big\|A^{1/2}/\|A\|^{1/2} - I + (I-A/\|A\|)/2 \big\| = 1/8.$$
Error bound using the third term: $$\big\|(I-A/\|A\|)^2/8 \big\| = 9/128.$$

[Remark: This is [cross posted][2].]


  [1]: https://en.wikipedia.org/wiki/Holomorphic_functional_calculus
  [2]: https://math.stackexchange.com/questions/3840411/error-bounds-on-the-expansion-of-square-root-of-matrix