Work out singular value decomposition instantly with clear inputs, formula shown and shareable results.
The singular values of A are the square roots of the eigenvalues of AᵀA. They measure how much the matrix stretches space in its principal directions, and their ratio is the condition number governing numerical stability.
Singular values
σᵏ = √(eigenvalue k of AᵀA)
Frobenius norm
‖A‖_F = √(Σσᵏ²) = √(Σaᵢⱼ²)
AᵀA = [[25, 23], [23, 26]] with eigenvalues 48.5054 and 2.4946, so σ₁ = 6.9646 and σ₂ = 1.5794 — and their product 11 equals |det A|.
The matrix is nearly singular, so small input errors get amplified enormously when solving with it.