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.
Formula
σᵢ = √λᵢ(AᵀA), σ₁ ≥ σ₂ ≥ … ≥ 0
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).
Frequently asked questions
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.