What is the minimizer of the matrix norm and it's significance?

Question

For $M_{n\times n}$ a p.s.d real matrix, if we minimize $||M^{\frac{1}{2}}x||_2$ over $x$ under a linear constraint on $x$ as in $Ax=b$, where $b$ is non-zero. what is the significance of this $x$? The matrix square root here is $U\lambda^{0.5}$ from its eigen-decomposition. Does it imply a notion of near-orthonormality to the eigen vectors in $M^{0.5}$? Am confused because $M$ contains the entire set of eigen vectors in it which are orthonormal themselves.