Singular Values Calculator — 2×2 Matrix SVD
Enter the four entries of a 2×2 real matrix to compute its singular values σ₁ ≥ σ₂ — the fundamental measures of how the matrix stretches vectors — plus the condition number, nuclear norm, and Frobenius norm.
Spectral norm — maximum stretch factor of the matrix
- 1
Frobenius norm² T = a² + b² + c² + d²
3² + 1² + 1² + 3² = 20Equals the trace of AᵀA — the sum of its diagonal entries. - 2
Determinant det(A) = ad − bc
3 × 3 − 1 × 1 = 8 - 3
Discriminant T² − 4 det(A)²
20² − 4 × 8² = 144 - 4
Largest eigenvalue λ₁ = (T + √disc) ÷ 2
(20 + 12) ÷ 2 = 16 - 5
Largest singular value σ₁ = √λ₁
√16 = 4
How does this calculator work?
Singular values σ₁ ≥ σ₂ of a 2×2 matrix A are √λ of AᵀA, found via the quadratic formula using trace ‖A‖²_F and det(A)². They measure the maximum (σ₁) and minimum (σ₂) stretch factors; their ratio is the condition number. σ₂ = 0 means the matrix is singular.
Formula
How this is calculated
The singular values of a real matrix A are the square roots of the eigenvalues of the product AᵀA, which is always symmetric and positive semi-definite (eigenvalues ≥ 0). For a 2×2 matrix with entries [a, b; c, d], the trace of AᵀA equals the squared Frobenius norm T = a²+b²+c²+d², and its determinant equals det(A)² = (ad−bc)². Applying the quadratic formula gives the two eigenvalues λ₁ ≥ λ₂ ≥ 0, and σᵢ = √λᵢ.
Singular values have a clear geometric meaning: the unit circle in ℝ² maps under A to an ellipse whose semi-axes have lengths σ₁ and σ₂. The largest singular value σ₁ is the matrix 2-norm (spectral norm) — the maximum stretch factor. The ratio σ₁/σ₂ is the condition number: a large condition number signals near-singularity, meaning small input changes can cause large output changes. When σ₂ = 0 the matrix is exactly singular (non-invertible) and the condition number is ∞.
This calculator computes σ₁ and σ₂ from first principles, then derives the nuclear norm (σ₁+σ₂), the Frobenius norm (√(σ₁²+σ₂²) = ‖A‖_F), and the condition number. The grid animation shows the unit square deforming under A, making the ellipse shape and stretch factors visible.
Frequently asked questions
The singular values of A are the square roots of the eigenvalues of AᵀA (or equivalently AAᵀ). For symmetric positive definite matrices, the singular values equal the absolute values of the eigenvalues, but in general they differ. Singular values are always non-negative.
A large condition number σ₁/σ₂ indicates the matrix is nearly singular. Linear systems Ax = b with large condition numbers are numerically ill-conditioned — small errors in the right-hand side b can produce large errors in the solution x. A matrix with σ₂ = 0 is exactly singular and has condition number ∞.
Yes — SVD generalises to any m×n matrix, yielding min(m, n) singular values. For matrices larger than 2×2, iterative numerical algorithms such as Golub–Reinsch are used. This calculator specialises in the 2×2 case where the closed-form quadratic formula gives exact values.
Also known as
TG we-Calculate Editorial Team. (2026). Singular Values Calculator — 2×2 Matrix SVD [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/singular-values-calculator
TG we-Calculate Editorial Team. "Singular Values Calculator — 2×2 Matrix SVD." TG we-Calculate. 2026. https://we-calculate.com/calculator/singular-values-calculator.
TG we-Calculate Editorial Team, "Singular Values Calculator — 2×2 Matrix SVD," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/singular-values-calculator
@misc{wecalculate_singular_values_calculator, title = {Singular Values Calculator — 2×2 Matrix SVD}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/singular-values-calculator}}, year = {2026}, note = {TG we-Calculate} }
Did this calculator help you?