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. 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.]

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