Calculation of canonical orthogonalized basis from overlap matrix?

From Equation [3.169] in David Cook's [Book](https://xn--webducation-dbb.com/wp-content/uploads/2020/11/Oxford-Science-Publications-David-B.-Cook-Handbook-of-computational-quantum-chemistry-Oxford-University-Press-USA-1998.pdf) , I understand that we use eigenvalues and eigenvectors of the overlap matrix to calculate an orthonormal matrix using the formula $$X=Us^{-1/2}$$, where $U$ is the matrix of eigenvectors and $s$ is the matrix of eigenvalues for the overlap matrix. But, when I tried to calculate the dot product between two vectors from the new matrix $X$, I do not get $\phi\_i\phi\_j=1$ or $0$ depending on if $i=j$ or $i\neq j$. Does this mean that I have to normalize $X$?




It was not a requirement that *X* had to be orthogonal. The only requirement was that $X^{\dagger}SX$ must be orthonormal.


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