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I can understand the concept of *eigenvectors are the vectors that will only scaled by the transformation T, such as Tx = $\lambda$x where x is the eigenvector.

However, when people are talking about PCA, they always say the eigenvectors of the covariance matrix are the principle components.

One is about transformation Tx = $\lambda$x, the other is about covariance matrix Cx = $\lambda$x. How do we see the covariance matrix as a transform, to fit the framework of my first *statement?

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    Too short for answer, but I think this might help you. Covariance matrix for sure has some geometric meaning. What I was told in our probability course is that if you have random vector $\mathbf{\xi}$ and you somehow decide to cook up a random scalar variable $v^T \mathbf{\xi}$, then variance of this random scalar variable is just $v^T \mathbf{C} v$, where $\mathbf{C}$ is your covariance matrix. The better analogy here is not a linear transform, but studying properties of quadratic form $x^T A x$ by means of eigenvalues and eigenvectors. – 2017-02-22

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