Connect SVD to PCA

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Problem

For the centered matrix

X=\begin{bmatrix}2&0\\0&1\\-2&0\\0&-1\end{bmatrix},

compute $X^{\mathsf T}X$ , the singular values, the first PCA direction, and the first-component explained-variance fraction.

Reveal answer or reference solution

The Gram matrix is

X^{\mathsf T}X=\begin{bmatrix}8&0\\0&2\end{bmatrix},

so the singular values are $\sqrt{8}=2\sqrt{2}$ and $\sqrt{2}$ . The right singular vector for the larger value is $(1,0)$ , up to sign, and is the first PCA direction. The explained fraction is

\frac{8}{8+2}=\frac{4}{5}.

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Connect SVD to PCA

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