SVD Calculator — 2x2 Matrix Singular Values
Singular value decomposition writes any matrix A as UΣVᵃ, where Σ holds the singular values on its diagonal. For a 2x2 matrix, the two singular values are the square roots of the eigenvalues of AᵃA — no need for the full U and V matrices to get just the sizes. Enter the four entries a, b, c, d to get σ₁ and σ₂.
From AᵃA to Singular Values
For A = [[a,b],[c,d]], AᵃA = [[a²+c², ab+cd],[ab+cd, b²+d²]]. Its trace is a²+b²+c²+d² and its determinant is (ad−bc)² — the square of A's own determinant. Solving λ² − trace·λ + det = 0 gives the two eigenvalues λ₁ ≥ λ₂ ≥ 0, and σ₁ = √λ₁, σ₂ = √λ₂.
Worked Example: A Diagonal Matrix
Take A = [[2,0],[0,3]]. Trace of AᵃA = 4+9 = 13, determinant = 6² = 36. The quadratic λ² − 13λ + 36 = 0 factors to (λ−9)(λ−4) = 0, giving λ₁=9, λ₂=4, so σ₁=3 and σ₂=2 — exactly the two diagonal entries, sorted largest first, which is always true for a diagonal matrix.
Why Singular Values Matter
σ₁ is the largest factor by which A can stretch a unit vector; σ₂ is the smallest. Their ratio σ₁/σ₂, the condition number, says how distorted a circle becomes after the transformation — a large ratio means A squashes space almost flat along one direction, a warning sign for numerical stability in anything built on top of that matrix.
Frequently Asked Questions
What is a singular value?
A singular value measures how much a matrix stretches space along a particular direction. For a 2x2 matrix there are exactly two, σ₁ ≥ σ₂ ≥ 0, found from the eigenvalues of AᵃA rather than of A itself - unlike eigenvalues, singular values are always real and non-negative, even for a non-square or non-symmetric matrix.
How do you compute singular values by hand for a 2x2 matrix?
Form AᵃA, take its trace (a²+b²+c²+d²) and determinant ((ad−bc)²), then solve the quadratic λ² − trace·λ + det = 0 for its two eigenvalues. Each singular value is the square root of one eigenvalue.
What's the difference between singular values and eigenvalues?
Eigenvalues only exist cleanly for square matrices and can be negative or complex. Singular values exist for any matrix, come from AᵃA (or AAᵃ), and are always real and non-negative - for a symmetric positive-definite matrix the two coincide, but in general they don't.
What does it mean if one singular value is 0?
The matrix is singular (non-invertible) and its determinant is 0 - since det(A) = ±σ₁σ₂, a zero singular value forces the determinant to zero too, meaning the matrix collapses a 2D input onto a line or point.
Why doesn't this calculator show U and V?
Finding the full decomposition requires computing orthonormal eigenvectors of AAᵃ and AᵃA, a heavier computation. This tool answers the more common quick question - how much does this matrix stretch space - which only needs the singular values themselves.