Raising a square matrix to a negative half power

Question

I want to implement the following formula (taken from Kaiser, 1970) in R where $R$ is square matrix of correlations:

$$S = (\textrm{diag } R^{-1})^{-1/2}$$

I understand the diagonal and inverse operations, but I am unclear on the meaning of raising a square matrix to a negative half power. Thus, my questions

1. What does it mean to raise a square matrix to a negative half power?
2. What general ideas of linear algebra does this assume?
3. (if this is in scope) How would this be implemented in R?

References

• Kaiser, H. F. (1970). A second generation little jiffy. Psychometrika, 35(4), 401-415.

Answer 1

If you look at Powers of diagonal matrices, it would be the reciprocal of the square root of each term in the diagonal.

Lets do an example for a diagonal matrix:

$$A=\begin{bmatrix}2 & 0 \\ 0 & 3\end{bmatrix}$$

$$A^{-1/2} = \begin{bmatrix} \frac{1}{\sqrt{2}} & 0 \\ 0 & \frac{1}{\sqrt{3}}\end{bmatrix} = \frac{1}{6}\begin{bmatrix}3 \sqrt{2} & 0 \\ 0 & 2 \sqrt{3}\end{bmatrix}$$

Clear?

Answer 2

The general idea (I think so) is Matrix function and Taylor expansion. Let $R$ is square matrix, we may write it in the form $R = I+X$ ($I$ - identity matrix), then using Taylor expansion: $$ R^{-1/2} = (I+X)^{-1/2} = I -\frac{1}{2}X + \frac{3}{8}X^2 + \cdots $$ I would be nice if $X$ is small.