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}.

Local history

Loading attempts saved in this browser…

Use with your agent

Share this URL and your attempt. Ask the agent to start with a clarifying question or the smallest useful hint.

Tutor me on https://mlprep.iwase.dev/mathematics/linear-algebra/original-la-low-rank/. If window.mlPrepAgent is available, read attempts for item original-la-low-rank before tutoring. Inspect my attempt, keep the item ID, and do not reveal the full answer first. After a real attempt, append its record and read it back.

Appears in

Linear Algebra Practice
Dimensionality Reduction Practice