For simplicity, let's consider a 2D case only.
I am measuring coordinates of say some 1000 data points lying on a regular 2D cartesian plane. The recorded $(x,y)$ values are placed as columns of matrix $X$ having dimensions $2 \times 1000$.
If I now attempt to get the covariance matrix $XX^\top$ mostly it will have several off diagonal entries. Then as popularised in approaches like PCA, we choose a basis vector, one of which is along the direction of maximum variance and the second one orthogonal to it. The eigen value for first eigenvector will be large (perhaps much larger) compared to the second eigenvalue.
Let the new system matrix using the new set of eigenvectors be $Y$ instead of $X$.
Now the condition number is ratio of the largest to smallest eigenvalue. Thus I expect that $X$ will have better condition number than $Y$.
Can I thus conclude that for solving equations $X$ should be preferred over $Y$, which was obtained using eigen-value-decomposition as in PCA (here though we are not doing any dimensionality reduction)?