TITLE: Raising a square matrix to a negative half power
QUESTION [12 upvotes]: 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

What does it mean to raise a square matrix to a negative half power?
What general ideas of linear algebra does this assume?
(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.

REPLY [2 votes]: So this goes along with Gr3gT's answer, but I'll chime in anyway.
Any $m \times n$ matrix $A$ can be written as:
$A = U \Sigma V^{H}$
Where $U$ is an $m\times m$ matrix whose columns are the left eigenvectors, $V$ is an $n\times n$ matrix whose columns are the right eigenvectors, and $\Sigma$ is a diagonal matrix of singular values. 
Since $U$ and $V$ are unitary, we have: 
$A^{\frac{1}{2}} = U \Sigma^{\frac{1}{2}} V^{H}$
So then:
$A^{\frac{-1}{2}} = (A^{\frac{1}{2}})^{-1} = (U \Sigma^{\frac{1}{2}} V^{H})^{-1}  = V \Sigma^{\frac{-1}{2}} U^{H}$
Now when $A$ is not square, we can simply compute:
$A^{\frac{-1}{2}} = (A^{+})^{\frac{1}{2}} = U_{+} \Sigma^{\frac{1}{2}}_{+} V^{H}_{+}$
By taking the SVD of $A^{+}$, the pseudo-inverse of $A$.