When optimizing a convex quadratic objective function, the positive definite matrix $\mathbf{Q}$ can be decomposed into its eigensystem as $\mathbf{Q} = \mathbf{O}^ op \boldsymbol{\Lambda} \mathbf{O}$, where $\mathbf{O}$ is an orthogonal matrix and $\boldsymbol{\Lambda}$ is a diagonal matrix of strictly positive eigenvalues. By applying the change of variables $\mathbf{z} = \mathbf{O} (\mathbf{x} - \mathbf{Q}^{-1} \mathbf{c})$, the optimization problem is recast into a much simpler coordinate system that aligns with the eigenvectors of $\mathbf{Q}$. A key theorem demonstrates that when expressed in this eigensystem, both standard gradient descent and gradient descent with momentum do not mix different eigenspaces. Instead, the multi-dimensional optimization problem decomposes entirely into independent, coordinate-wise optimization processes along the directions of the eigenvectors.