[Math] How to calculate the negative half power of a matrix

Question

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

Answer 1

$A=SDS^{−1}$, 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 $A^{-1/2}=(A^{1/2})^{-1}$