Singular Value Decomposition Calculator

The singular value decomposition factors any matrix A into A = U Σ V^T, where:

• U is orthogonal — its columns (left singular vectors) form an orthonormal basis
• Σ is diagonal with non-negative entries (singular values), conventionally in decreasing order
• V is orthogonal — its columns (right singular vectors) form an orthonormal basis

The singular values σ₁ ≥ σ₂ ≥ 0 measure how much the matrix stretches space along the corresponding right singular vectors. Geometrically: the unit circle is mapped to an ellipse with semi-axes σ₁ and σ₂.

For a 2x2 matrix this calculator solves analytically. The squared singular values are the eigenvalues of A^T A:

A^T A = [[a²+c², ab+cd], [ab+cd, b²+d²]]

Eigenvalues of this symmetric 2x2 matrix come from the quadratic formula on its characteristic polynomial. Then σᵢ = √λᵢ.

The right singular vectors (columns of V) are the eigenvectors of A^T A. The left singular vectors (columns of U) come from u_i = A v_i / σ_i. When σ_i = 0, the corresponding u_i can be any unit vector orthogonal to the others.

Worked example: A = [[3,0],[4,5]]. A^T A = [[25, 20],[20, 25]]. Eigenvalues: trace = 50, det = 625 - 400 = 225. λ = (50 ± √(2500 - 900))/2 = (50 ± 40)/2 = 45 or 5. Singular values: σ₁ = √45 ≈ 6.708, σ₂ = √5 ≈ 2.236.

The condition number κ(A) = σ_max / σ_min measures how badly small input changes amplify into output changes. For this example: κ = √45 / √5 = 3. Larger condition numbers signal a near-singular matrix where numerical methods struggle.

The SVD has a vast range of applications: principal component analysis (PCA) starts here, low-rank matrix approximations for data compression and recommender systems, the Moore-Penrose pseudoinverse, stable least-squares solving, and image compression all rest on it. For matrices larger than 2x2, the analytical approach breaks down — practical SVD uses iterative algorithms like one-sided Jacobi rotations or QR-based methods.