How to calculate the negative half power of a matrix 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
 A: $A=SDS^{−1}$, and $A^k=SD^kS^{−1}$ where D is the diagonal matrix composed from eigenvalues, S is the matrix of eigenvectors, and $S^{−1}$ is the inverse of S.
So the first step is to find the eigenvalues, and then find the corresponding eigenvectors.
Then use $A^{1/2}=SD^{1/2}S^{-1}$, and finally find the inverse $A^{-1/2}=(A^{1/2})^{-1}$
A: 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.
A: It exists only if $A$ is invertible, i.e. $A^{-1}$ exists. Now this matrix have to follow certain properties , then a square root can be computed.
You always have the EVD at hand. Break the matrix $A^{-1}=U\cdot \Sigma\cdot U^t$ Where $U$ is a unitary matrix.Check that $$(U\cdot \Sigma^{1/2}\cdot U^t)^2=U\cdot \Sigma^{1/2}\cdot U^t\cdot U\cdot \Sigma^{1/2}\cdot U^t=A^{-1}$$
And $[\Sigma^k]_{ii}=[\Sigma_{ii}^k]$
A: Eigen-decomposition: $A = U * \Lambda * U^t$.
$A^{-1/2} = U * \Lambda^{-1/2} * U^t$.
This method is not a very good for numerical computation.
