Ah I see. As I understand it a general linear map like that isn't what the linked paper means by "scale-invariance", so it wouldn't be considered a violation for a dataset and it's PCA to be given different clusters by your clustering algorithm. It's only the dataset and its scaled up or down counterparts (i.e. the metric is multiplied by a fixed non-zero constant) that are required to get the same clusters for scale-invariance to hold.
In fact the paper doesn't assume that your dataset is contained in a vector space at all. All you have to give a clustering algorithm (as they define it) is a set and a metric function on it.
In fact the paper doesn't assume that your dataset is contained in a vector space at all. All you have to give a clustering algorithm (as they define it) is a set and a metric function on it.
(the paper if you don't have a link: https://www.cs.cornell.edu/home/kleinber/nips15.pdf)