---
tag: 0K2TX12X58
kind: definition
name: Singular value
msc: [15A18]
defines: 09VMDSRXWJ
uses: [0AM050KCAW, 0ZND7NQ970]
sources:
  - url: https://en.wikipedia.org/wiki/Singular_value
    locator: lead section
---

## Definition

Let $A$ be an $m \times n$ complex matrix.
The matrix $A^* A$ is positive semidefinite, so its [[eigenvalues|0BQAA2BK3Q]] $\lambda_1 \ge \cdots \ge \lambda_n \ge 0$ are real and nonnegative.
The *singular values* of $A$ are the numbers
$$\sigma_i = \sqrt{\lambda_i} \qquad (1 \le i \le n).$$

## Remarks

The number of nonzero singular values is the rank of $A$.
The largest singular value is the operator norm $\max_{\norm{x} = 1} \norm{Ax}$.
For a normal matrix, the singular values are the absolute values of the eigenvalues.
Some authors list only the first $\min(m, n)$ values.

## Explanations

### L1 human

The singular values of a matrix measure how much it stretches space in its main directions.
The unit sphere goes to an ellipsoid, and the singular values are the lengths of its semi-axes.
Unlike eigenvalues, they exist for every matrix, also rectangular ones.