[Math] Raising a square matrix to a negative half power

exponentiationlinear algebramatrices

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.

Best Answer

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?

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