---
title: "The Singular Value Decomposition"
description: "Every matrix, square or not, factors into a rotation, a scaling and another rotation. The singular values rank the directions by how much the map stretches them, which is how low-rank approximation wo"
canonical: https://lightmysky.com/learn/mathematics/the-singular-value-decomposition-mt_UsUZbM9VRK
source: https://lightmysky.com/learn/mathematics/the-singular-value-decomposition-mt_UsUZbM9VRK.md
retrieved: 2026-09-12
---

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# The Singular Value Decomposition

Every matrix, square or not, factors into a rotation, a scaling and another rotation. The singular values rank the directions by how much the map stretches them, which is how low-rank approximation works.

Subject: Mathematics · Area: Linear Algebra · Ages 20 to 21
Page: https://lightmysky.com/learn/mathematics/the-singular-value-decomposition-mt_UsUZbM9VRK

## Ready when they can

- Relate singular values to eigenvalues of the matrix times its transpose
- Read the rank of a matrix off its singular values
- Explain how truncating the decomposition gives a best low-rank approximation

## Lesson: Stretch directions, ranked by size

Every matrix, square or not, factors into a rotation, a scaling, and another rotation. That is the singular value decomposition. The scalings are the singular values, and they rank the directions by how much the map stretches them. The largest singular value tells you the biggest stretch factor the map applies to any direction.

**Example.** Take the diagonal matrix with rows (3, 0) and (0, 4). Form A transpose times A, which is diagonal with 9 and 16. The eigenvalues are 9 and 16, and their square roots are 3 and 4, so the largest singular value is 4. A wilder one: rows (3, 4) and (0, 0) give A transpose times A with trace 25 and determinant 0, hence eigenvalues 25 and 0, so the largest singular value is 5.

The recipe generalises: singular values are the square roots of the eigenvalues of A transpose times A, which are always real and nonnegative. Because of that link, you can read the rank straight off the singular values: count the nonzero ones. A zero singular value marks a direction the map flattens completely. Keep your transpose fluent, since one sign slip there ruins the whole chain.

**Tip.** Truncation is the payoff. Dropping the small singular values and keeping the big ones gives the best low-rank approximation of the matrix, which is how compression works. Review rank and symmetric diagonalisation first, and the whole decomposition reads as those two ideas combined.

**Recap.** Square root the eigenvalues of A transpose times A to rank every stretch direction of the map.

## Practice

14 questions on this page, each with its working shown.

## Needs first

- [Orthogonal Projection and Least Squares](https://lightmysky.com/learn/mathematics/orthogonal-projection-and-least-squares-mt_kGvRrYJrat)
- [Symmetric Matrices and the Spectral Theorem](https://lightmysky.com/learn/mathematics/symmetric-matrices-and-the-spectral-theorem-mt_sdQv4m7Nbk)

## Opens up

- [Principal Components and Dimensionality Reduction](https://lightmysky.com/learn/computing/principal-components-and-dimensionality-reduction-mt_l-9XHg_ChZ)