SVD, best rank one, and PCA energy

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Problem

Let $X=\operatorname{diag}(4,2,1)$ . Give one SVD of $X$ , its best rank- $1$ approximation in Frobenius norm, and the approximation error. If $4,2,1$ are the singular values of a centered data matrix, what fraction of its total squared variation is retained by the first principal component?

Reveal answer or reference solution

Since $X$ is diagonal with positive diagonal entries, one SVD is $X=I\operatorname{diag}(4,2,1)I^{\mathsf T}$ . By Eckart-Young, retain only the largest singular value:

X_1=\operatorname{diag}(4,0,0)

The Frobenius error is

\|X-X_1\|_F=\sqrt{2^2+1^2}=\sqrt5

For a centered data matrix with these singular values, the first principal direction is the first right singular vector $e_1$ , and the retained squared variation is

\frac{4^2}{4^2+2^2+1^2}=\frac{16}{21}

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SVD, best rank one, and PCA energy

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