# Singular Values Calculator

URL: https://speedycalculator.com/calculators/math/singular-values/
Last updated: 2026-10-30

> The singular values calculator finds the singular values of a 2 × 2, 2 × 3, 3 × 2 or 3 × 3 matrix – the square roots of the eigenvalues of AᵀA – and the 2-norm condition number.

## Inputs

- Matrix size (rows × columns)
- a₁
- b₁
- c₁
- a₂
- b₂
- c₂
- a₃
- b₃
- c₃

## Results

- 1st singular value σ₁
- 2nd singular value σ₂
- 3rd singular value σ₃
- Condition number σ₁/σₙ

## Formula

```
σᵢ = √λᵢ(AᵀA),  σ₁ ≥ σ₂ ≥ … ≥ 0
```

## Explanation

Singular values measure how much a matrix stretches space in each direction. They give the rank, the 2-norm (σ₁), the condition number (σ₁/σₙ) and the best low-rank approximations used in image compression and recommender systems.

### How to use the Singular Values Calculator

- Choose the **matrix size** – the matching grid of fields appears.
- Enter the entries **row by row** (a₁ b₁ c₁ is the first row, a₂ b₂ c₂ the second, and so on).
- Read the result; integer matrices give exact fractions.

Everything is calculated in your browser, so results appear instantly and nothing you type is sent to a server.

### How it works

AᵀA is symmetric, so its eigenvalues are found with the Jacobi method; their square roots, in decreasing order, are the singular values.

## Worked example

Inputs — Matrix size (rows × columns): 2 × 2; a₁: 3; b₁: 0; a₂: 4; b₂: 5.

Results — **1st singular value σ₁ = 6.708203932**; 2nd singular value σ₂ = 2.236067977; Condition number σ₁/σₙ = 3.

## Key terms

- **Singular value**: √(eigenvalue of AᵀA).

## FAQ

**Are singular values always non-negative?**
Yes.

**How many singular values are there?**
min(m, n) for an m × n matrix.

**What do zero singular values mean?**
The matrix is rank-deficient.

**Are singular values the same as eigenvalues?**
Only for symmetric positive semi-definite matrices.


## Sources

- [Wolfram MathWorld – Singular value](https://mathworld.wolfram.com/SingularValue.html)
- [MIT OpenCourseWare – 18.06 Linear Algebra (Gilbert Strang)](https://ocw.mit.edu/courses/18-06-linear-algebra-spring-2010/)
- [Jim Hefferon – Linear Algebra (free textbook)](https://hefferon.net/linearalgebra/)