Singular Value Decomposition Calculator

Work out singular value decomposition instantly with clear inputs, formula shown and shareable results.

About Singular Value Decomposition Calculator

The singular values of A are the square roots of the eigenvalues of AᵀA. They measure how much the matrix stretches space in its principal directions, and their ratio is the condition number governing numerical stability.

Formulas

Singular values

σᵏ = √(eigenvalue k of AᵀA)

Frobenius norm

‖A‖_F = √(Σσᵏ²) = √(Σaᵢⱼ²)

Worked example

With the default values below, Singular Value Decomposition Calculator returns Largest singular value σ₁ of 6.96458428. Change any input above and every figure here is recalculated in your browser.

Inputs used

Entry a: 3
Entry b: 1
Entry c: 4
Entry d: 5

Results

Largest singular value σ₁: 6.96458428
Smallest singular value σ₂: 1.57941947
Condition number σ₁/σ₂: 4.40958492
Frobenius norm: 7.14142843

Step by step

1Values used
2Singular values
3Frobenius norm
4Largest singular value σ₁= 6.96458428
5Smallest singular value σ₂= 1.57941947
6Condition number σ₁/σ₂= 4.40958492
7Frobenius norm= 7.14142843

Assumptions and limits

The result is derived only from the 4 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 8 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

For [[3,1],[4,5]]?

AᵀA = [[25, 23], [23, 26]] with eigenvalues 48.5054 and 2.4946, so σ₁ = 6.9646 and σ₂ = 1.5794 — and their product 11 equals |det A|.

What does a large condition number mean?

The matrix is nearly singular, so small input errors get amplified enormously when solving with it.