I am looking for an asymptotic expansion for $\underline\gamma$ which is the "square root" matrix of a symmetric $p\times p$ matrix $\gamma$. Here $\underline\gamma$ is assumed to be symmetric, e.g. obtained by singular value decomposition. Specifically, I need the series of the square root matrix to be developed from that of $\gamma$ (and the latter series is in powers of $n^{-1/2}$, where $n$ is some asymptotic index, to be not discussed here as it is not relevant for the discussion).
By making use of index notation let $\gamma_{rs}$ and $\underline\gamma_{rs}$ denote the entries of the matrices as $r,s$ range in $1,\dots, p$. In the following summation convention is understood.
Here it is what I was able to develop so far: I was able to find the expansion for $\gamma_{rs}$ which is of the form $\gamma_{rs}+n^{-1/2}c_{rs}$. Now to obtain that for $\underline\gamma_{rs}$ I started from the identity $\gamma_{rs}=\underline\gamma_{ra}\underline\gamma_{as}$ and, by assuming that $\underline\gamma_{rs}$ admits an expansion of the form $\underline\gamma_{rs}+n^{-1/2}\underline c_{rs}$, I am able to set the equation $\underline\gamma_{ra}\underline c_{as}+\underline\gamma_{as}\underline c_{ra}=c_{rs}$. However, the tricky part is that $\underline c_{rs}$ is easily seen to be NON symmetric under permutation of the indexes, i.e. $\underline c_{rs}\neq \underline c_{sr}$. So, how it is possible to obtain the expansion of the "square root" matrix $\underline\gamma$ from that of $\gamma$?
Thank you in advance,
violator
