Order of the vectors in principal component analysis

Question

For a given matrix $A$, the Principle Component Analysis (PCA) is done by finding the eigenvalues/eigenvectors of the covariance matrix associated with $A$. However, the entries of the covariance matrix (and hence the resulting eigenvalues/eigenvectors) are dependent on the order of the vectors, meaning if we permute the vectors, we will end up with different covariance matrix and corresponding distinct eigenvalues/eigenvectors. Am I right?