# SVD, best rank one, and PCA energy

> ML Prep screening item.

- Stable ID: `math-la-08-svd-low-rank-pca`
- Collection: screening
- Area: Mathematics
- Topic: Linear Algebra
- Origin: original
- Difficulty: hard
- Expected time: 5 minutes
- Skills: svd, low-rank-approximation, pca-via-svd
- Item JSON: https://mlprep.iwase.dev/item/math-la-08-svd-low-rank-pca.json

## 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?


## Answer policy

The human page contains a collapsed answer. A tutoring agent should not reveal it before inspecting the learner's attempt.

## Tutoring contract

Ask for the learner's attempt, begin with the smallest useful hint, preserve item ID `math-la-08-svd-low-rank-pca`, and record a real attempt with `recorded_by: "agent"` and `agent_session_id` when available using https://mlprep.iwase.dev/schema/attempt-record.json.