Projections in new coordinates in PCA example

Question

I am currently studying principal component analysis in statistics. I am presented with the following example:

$\mathbf{X} = [X_1 \ X_2]^T$ a two-dimensional random vector with mean $\mathbf{\mu} = [0, -1]^T$, covariance matrix $\Sigma$, and eigenvalue-eigenvector pairs $(\lambda_j, \mathbf{\eta}_j), j = 1, 2$ given by $$\Sigma = \begin{bmatrix} 2.4 & -0.5 \\ -0.5 & 1 \end{bmatrix}, (\lambda_1, \mathbf{\eta}_1) = \left( 2.56, \begin{bmatrix} 0.95 \\ -0.31 \end{bmatrix} \right), (\lambda_2, \mathbf{\eta}_2) = \left( 0.84, \begin{bmatrix} 0.31 \\ 0.95 \end{bmatrix} \right)$$ Rotate the coordinate system: the first eigenvector agrees with the $x$-axis and the second with the $y$-axis.
Calculate projections in the new coordinates, new values $W_1$, $W_2$: $$W_1 = 0.95X_1 − 0.31(X_2 + 1) \ \ \ \text{and} \ \ \ W_2 = 0.31X_1 + 0.95(X_2 + 1)$$

I have some confusion with this example. Where did these calculations $W_1 = 0.95X_1 − 0.31(X_2 + 1) \ \ \ \text{and} \ \ \ W_2 = 0.31X_1 + 0.95(X_2 + 1)$ come from? In particular, where did the $(X_2 + 1)$ come from, whereas $X_1$ was chosen for the first term? Also what is meant by "the first eigenvector agrees with the $x$-axis and the second with the $y$-axis"?

Answer 1

We are subtracting from the mean, hence shifting the mean as the new origin.

We are told the mean is $[0, -1]$.

Hence we first subtract $0$ from $X_1$ and we subtract $-1$ from $X_2$. Hence that is how we obtain $(X_1, X_2+1)$.

After that we project it to the eigenvector, $(0.95, -0.31)$, this is done by taking the inner product of $(X_1, X_2+1)$ and $(0.95, -0.31)$.

Similarly for the second eigenvector.