A data set $S \subset \Bbb R^p$ consists of vectors whose first component is the number of kilometers that a salesperson has travelled during the last month. A principal component analysis is performed on $S$, the set $S$ is then projected onto the three principal components with largest eigenvalues, and the projected points are visualized in $\Bbb R^3$ . Would this visualization be any different if distance had been measured in miles?
I understand that the visualisation would be different as the three largest eigenvalues would be translated but cannot understand entirely why.
If the rest of the dataset remains in kilometers, then certainly the three principle components can change quite drastically. It's generally not recommended to use PCA when the data points are measured in different units unless you have no other choice because of this reason (and the result may not be particularly enlightnening).
But if you change all of the units to miles, then you shouldn't see a change in visualization other than the shrinking/expanding associated with changing units. The shape of the visualization should remain the same.