power series expansion of the square root of a Hermitian matrix

Is there a power series development of the square origin of a Hermitian matrix, as a procedure to compute the square origin without taking the inverted or diagonalizing the matrix? I locate for scalar number $x$, $$\sqrt{x}=\sum_{k=0}^\infty \frac{(-1)^k \left((-1+x)^k \left(-\frac12\right)_k\right)}{k!}\qquad\text{for }|-1+x|<1$$, under what problem can I make use of the very same development for a matrix?

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2022-07-25 20:46:47
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