SVD Calculator — Singular Value Decomposition (2×2)
Enter a 2×2 matrix and instantly get its singular values σ₁ ≥ σ₂, condition number, rank, Frobenius norm and spectral norm — all from its singular value decomposition.
Square root of the largest eigenvalue of AᵀA — equals the spectral (2-)norm of A
- 1
AᵀA trace (= ‖A‖²_F)
3² + 1² + 1² + 2² = 15Sum of squares of all matrix entries; also the sum of both eigenvalues. - 2
AᵀA determinant
(10) × (5) − (5)² = 25 - 3
Дискримінант
15² − 4 × 25 = 125 - 4
Largest singular value σ₁
√((15 + √125) ÷ 2) = 3,6180
Як працює цей калькулятор?
Form AᵀA (trace T = a²+b²+c²+d², det D = (ad−bc)²), solve λ²−Tλ+D = 0, take σᵢ = √λᵢ. Condition number κ = σ₁/σ₂; Frobenius norm = √T; spectral norm = σ₁; rank = count of σ > 0. For a 2×2 matrix, SVD measures how much the matrix stretches and squishes space along its two principal directions.
Формула
How this is calculated
Singular Value Decomposition (SVD) writes any real matrix A as A = UΣVᵀ, where U and V are orthogonal matrices and Σ is a diagonal matrix of non-negative singular values σ₁ ≥ σ₂ ≥ 0. The singular values are the square roots of the eigenvalues of the symmetric positive semi-definite matrix AᵀA.
For a 2×2 matrix A = [[a, b], [c, d]], this calculator forms AᵀA, extracts its trace T = a²+b²+c²+d² and determinant D = (ad−bc)², then solves the characteristic equation λ²−Tλ+D = 0 via the quadratic formula to get the two eigenvalues. Their square roots give σ₁ and σ₂. The condition number κ = σ₁/σ₂ measures numerical sensitivity: κ = 1 means perfectly conditioned; a large κ (> 100) means near-singular and numerically ill-behaved. Rank equals the count of positive singular values.
The Frobenius norm ||A||_F = √(a²+b²+c²+d²) = √(σ₁²+σ₂²) and spectral norm ||A||₂ = σ₁ are also returned. This calculator computes only the scalar σ values — the full rotation matrices U and V are not shown.
Поширені запитання
They are the semi-axis lengths of the ellipse that A maps the unit circle onto. σ₁ is the longest axis (maximum stretch), σ₂ the shortest. The product σ₁·σ₂ = |det(A)| is the area-scaling factor of the transformation.
A high condition number κ = σ₁/σ₂ (typically > 100–1000) means small input perturbations cause disproportionately large output errors — numerical computations become unreliable. A singular matrix has σ₂ = 0 and κ = ∞.
Eigenvalues of a non-symmetric A can be complex, while the singular values are always real and non-negative by definition. AᵀA is symmetric positive semi-definite, so its eigenvalues are guaranteed to be real and ≥ 0, making their square roots well-defined.
Також відомий як
TG we-Calculate Editorial Team. (2026). SVD Calculator — Singular Value Decomposition (2×2) [Online calculator]. TG we-Calculate. https://we-calculate.com/uk/calculator/svd-calculator
TG we-Calculate Editorial Team. "SVD Calculator — Singular Value Decomposition (2×2)." TG we-Calculate. 2026. https://we-calculate.com/uk/calculator/svd-calculator.
TG we-Calculate Editorial Team, "SVD Calculator — Singular Value Decomposition (2×2)," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/uk/calculator/svd-calculator
@misc{wecalculate_svd_calculator, title = {SVD Calculator — Singular Value Decomposition (2×2)}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/uk/calculator/svd-calculator}}, year = {2026}, note = {TG we-Calculate} }
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