Low-rank error and explained variance

Problem

A centered data matrix has singular values $5$ , $2$ , and $1$ . Find the Frobenius error of its best rank- $1$ approximation and the fraction of total squared variation retained by the first principal component.

Reveal answer or reference solution

The best rank- $1$ error is

\sqrt{2^2 + 1^2} = \sqrt{5}.

The retained fraction is

\frac{5^2}{5^2 + 2^2 + 1^2} = \frac{25}{30} = \frac{5}{6}.

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Appears in

Linear Algebra Practice
Dimensionality Reduction Practice