Step 1 Center

Subtract (5, 4) from each point

Step 2 Covariance matrix

• top-left = 10 → how much &x& spreads on its own
• bottom-right = 5 → how much $𝑦$ spreads on its own ($𝑥$ spreads twice as much)
• off-diagonal = 6 → how $𝑥$ and $𝑦$ move together (positive, so they rise together)

Step 3 — Eigenvalues and eigenvectors

Each principal component is an eigenvector $𝑣$ of $𝐶$: a direction that $𝐶$ only stretches, never rotates.

Eigenvalues — force a non-trivial solution with $det\left(𝐶-𝜆𝐼\right)=0$:

Eigenvectors — substitute each $𝜆$ back into $\left(𝐶-𝜆𝐼\right)𝑣=0$.

For ${𝜆}_1=14$, with $𝐶-14𝐼=\left[\begin{array}{cc}-4& 6\\ 6& -9\end{array}\right]$:

For ${𝜆}_2=1$, with $𝐶-1𝐼=\left[\begin{array}{cc}9& 6\\ 6& 4\end{array}\right]$:

Normalise to unit length: