# Projections in new coordinates in PCA example

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"?

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 $$(0.95, -0.31)$$.
• Thanks! Any idea what "the first eigenvector agrees with the $x$-axis and the second with the $y$-axis" means? Feb 15, 2021 at 4:41
• it's just a convention. after we map it, we can plot the result. so we plot $W_2$ vs $W_1$. , where $W_1$ is on the horizontal axis and $W_2$ on the vertical axis. Feb 15, 2021 at 4:49