computation of square root of matrix I am studying the Canonical correlation analysis(CCA)
the formula https://en.wikipedia.org/wiki/Canonical_correlation
is involving the -1/2 degree of a matrix.
My question is:
how  can i do the computation by using programme? Or in practice no one will actually care about the computation, since statistical programme ,like sas (proc cancorr ),will have the built in function on it
 A: Calculating the inverse square root of a square matrix $K$ is a fairly straight-forward process mathematically given that the matrix $K$ is a valid covariance matrix, ie. it is symmetric positive definite. This is the case in the calculation of canonical correlation as you concerned with the matrices $\Sigma_{YX}$ and $\Sigma_{XX}$ that are defined to be covariances matrices (more specifically  $\Sigma_{YX}$ and $\Sigma_{XX}$ are cross- and auto-covariance matrices respectively). Assuming a matrix $X$ such that it has the eigendecomposition $K = VDV^T$ where $V$ are the eigenvectors of it and $D$ the diagonal matrix holding the eigenvalues associated with the eigenvectors $V$, the inverse square root of a matrix $K$ is simply $K^{-\frac{1}{2}} = V D^{-\frac{1}{2}} V^T$. Taking the square-root and then inverting the elements of $D$ is trivial.
As you correctly note, a lot of statistical programs implement higher-level functions (eg. cancorr) so users does not need to compute a matrix inverse square root explicitly on their own. In many cases a higher-level routine is optimised by its developers in terms of speed and numerical accuracy.  If you want a particular higher-level routine I would recommend you use it directly as to ensure the correct of that computation. If you want to investigate this routine in-depth though, implementing it from scratch will be invaluable.
