When a quadratic optimization problem $f(\mathbf{x}) = \frac{1}{2} \mathbf{x}^\top \mathbf{Q} \mathbf{x} + \mathbf{c}^\top \mathbf{x} + b$ is simplified using the eigendecomposition $\mathbf{Q} = \mathbf{U}^\top \boldsymbol{\Lambda} \mathbf{U}$, the transformed variables become $\bar{\mathbf{x}} = \mathbf{U} \mathbf{x}$ and $\bar{\mathbf{c}} = \mathbf{U} \mathbf{c}$. In this simplified coordinate system, the optimal minimizer is directly computed as $\bar{\mathbf{x}} = -\boldsymbol{\Lambda}^{-1} \bar{\mathbf{c}}$, yielding a minimum function value of $-\frac{1}{2} \bar{\mathbf{c}}^\top \boldsymbol{\Lambda}^{-1} \bar{\mathbf{c}} + b$. This formulation is highly efficient to calculate because $\boldsymbol{\Lambda}$ is a diagonal matrix containing the eigenvalues of $\mathbf{Q}$, allowing each coordinate to be solved individually.