# Eigenvectors of covariance matrix vs Eigenvectors of transformation matrix and PCA

As far as I understand, if I multiply some normally distributed data by a transformation matrix T , data will be have maximum spread in the direction of the Eigen-Vector of the maximum Eigen-Value of this transformation matrix. Also we can see that the covariance matrix of the data can be seen as C = TT' , where (') is the transpose as seen here : https://robotics.stackexchange.com/questions/2556/how-to-rotate-covariance and here : http://www.lucidarme.me/?p=946 Now the PCA tells us that the direction of the maximum spread of Data will turn out to be the direction of the eigen-vector of the maximum eigenvalue of the covariance matrix itself.

Does this mean that the eigen-vectors of the covariance matrix C are the same as the eigen-vectors of T ? Can we prove this in a general case where T = RS , where S is scale component and R is a rotation component ? Or what am I doing wrong here ? I have read somewhere that matrix A has the same eigen-vectors as A'A iff A'A = AA' , but that is not the case for a general T = RS.