TITLE: How to calculate the negative half power of a matrix
QUESTION [5 upvotes]: I have a square matrix called A. How can I find $A ^ {-1/2}$. Should I compute $a_{ij} ^ {-1/2}$ for all of its elements?
Thanks

REPLY [2 votes]: Firstly you will need to pick a branch of the square root function over the field of which the elements of your matrix belong. Once you have done that,

If $A$ is diagonalizable you can do as @Annalise writes.
If $A$ is orthogonally diagonalizable you can do as @Junning Li writes.

However for example if $A$ is not diagonalizable, then it can maybe still be put on some canonical form:
$$A = TCT^{-1}$$
Where $C$ can be block-diagonal matrix. In this case we can do approximate square root by trying some power series expansion on the diagonal blocks of $C$. However this is not guaranteed to make any sense. It will depend a lot on application.