How this instrument works
Singular values measure how much a linear transformation stretches space along its most-amplified and least-amplified directions — unlike eigenvalues, which only exist for square arrays with well-behaved directions, singular values are always real and non-negative, for absolutely any such array at all. They're found as the square roots of the eigenvalues of AᵀA, which is always symmetric, guaranteeing those eigenvalues are always real and non-negative to begin with.
This page reuses this site's own Diagonalize Matrix machinery under the hood — computing AᵀA's trace and determinant, then solving the identical quadratic-formula eigenvalue equation that page already demonstrates, just applied to this new derived array rather than the original one directly.
For an invertible, already-symmetric starting array, the singular values and the (absolute values of the) eigenvalues coincide exactly — the extra generality singular values offer only becomes visible where ordinary eigenvalues behave badly or don't exist in a useful real form at all.
- Enter the matrix's four entries into the a, b, c, and d fields.
- Read Larger singular value: the bigger of the two stretch factors.
- Read Smaller singular value: the smaller of the two.
- Multiply the two singular values together and compare against the absolute value of the matrix's own determinant — they should match.
Worked example — [[1,1],[0,1]]
For [[1,1],[0,1]] — the same array this site's QR Decomposition page also uses as its own worked example — the singular values come out to the golden ratio, φ≈1.618, and its reciprocal, 1⁄φ≈0.618. Their product, φ×(1⁄φ)=1, matches its own determinant (1×1−1×0=1) exactly, as it always must.
The identity array [[1,0],[0,1]] stretches every direction identically, by exactly 1 — both singular values equal 1. A diagonal one like [[2,0],[0,3]] has singular values equal to the absolute values of its own diagonal entries, 2 and 3, listed largest first.
Questions
What is a singular value?
A measure of how much a linear transformation stretches space along one of its own special directions — every such array, square or not, invertible or not, has a full set of well-defined, real, non-negative singular values.
How are singular values different from eigenvalues?
Eigenvalues only exist in a simple real form for certain square arrays, and can be negative or complex; singular values always exist, are always real, and are always non-negative, for any array at all — found instead from the eigenvalues of AᵀA, which is always symmetric.
What does the product of the two singular values tell you?
For a 2×2 array, it always equals the absolute value of its own determinant — a quick way to double-check both singular values were computed correctly.
How is this different from the Diagonalize Matrix calculator on this site?
That page finds the eigenvalues of the array itself; this page finds the eigenvalues of a DIFFERENT, derived array, AᵀA, then takes their square roots — the underlying quadratic-formula method is identical, just applied to a different starting point.
What are the singular values of a diagonal array?
Simply the absolute values of its own diagonal entries — no further calculation needed, since such an array already stretches each axis independently by exactly that amount.