I'm having a tough time understanding the last "step" of Principal Component Analysis. Specifically, I'm trying to understand the following.
$$Z^* = ZP^*$$
In the step above, you take the standardized form, $Z$, of your original data and multiply it by $P^*$ where the columns of $P^*$ are the eigenvectors $v_1, v_2, ..., v_k $ of the covariance matrix arranged from largest corresponding eigenvalue to smallest.
Specifically, I'm having difficulty intuitively understanding how taking a row $z_i$ (or vector of input data) from $Z$ and multiplying it by the eigenvectors of the covariance matrix give you a new vector $z_i^*$ with values that correspond to the "weights" in the direction of each eigenvector.