How this instrument works
Picture a matrix acting on a perfect circle of input directions. Every matrix bends that circle into an ellipse, and the ellipse's long and short semi-axes are exactly its two singular values, σ₁ and σ₂. The condition number computed here is their ratio, σ₁ ⁄ σ₂ — a single figure describing how squashed that ellipse becomes. A ratio near 1 means the circle barely changes shape; a large ratio means one direction gets stretched far more than another, warping the circle into a thin, needle-like sliver.
This page leans on the same singular-value arithmetic behind this site's own Singular Values calculator, which reports σ₁ and σ₂ as two separate figures. Here, instead of listing both stretch amounts on their own, the sheet simply divides one by the other, folding two numbers into a single sensitivity score.
A separate page on this site, Condition Number, computes a figure with the identical name using an entirely different rule: the infinity norm, built from the largest absolute row sum of a matrix and of its inverse. That definition and this one are both legitimately called 'the condition number,' yet they measure different things and rarely agree on an exact value for the same four entries. The singular-value version used on this page — sometimes called the 2-norm or spectral condition number — is the definition most often reached for in numerical linear algebra, precisely because it ties directly to how a matrix reshapes a circle into an ellipse rather than to row-by-row arithmetic.
Either way the number is built, the reading works the same: near 1 signals a well-behaved matrix where small input errors stay small on the way out, while a large figure warns that inverting the matrix, or solving a linear system built from it, can turn tiny rounding noise into a wildly different answer. The identity matrix sits at the floor, exactly 1, since it does not favor any direction over another at all.
- Enter the matrix's four entries into a (row 1, col 1), b (row 1, col 2), c (row 2, col 1), and d (row 2, col 2).
- Read Condition number for σ₁ ⁄ σ₂, the largest singular value divided by the smallest.
- Try a=1, b=0, c=0, d=1 first to confirm the identity matrix returns exactly 1, the best possible score.
- Feed in a diagonal matrix like a=2, d=3 (b=c=0) to see the ratio read straight off the two diagonal entries.
- Cross-check against this site's Singular Values page, which reports σ₁ and σ₂ individually rather than as one ratio.
Worked example — three contrasting matrices
For A = [[1,1],[0,1]], the singular values are the golden ratio, φ ≈ 1.618033988749895, and its reciprocal, 1⁄φ ≈ 0.618033988749895. Dividing gives φ ⁄ (1⁄φ) = φ² ≈ 2.618033988749895, exactly the figure this calculator returns for these four entries.
A diagonal matrix keeps the arithmetic simple: for [[2,0],[0,3]], the singular values are just the diagonal entries themselves, 3 and 2, so the ratio is 3 ÷ 2 = 1.5. The identity matrix, [[1,0],[0,1]], stretches every direction by exactly the same amount, giving singular values of 1 and 1 and a condition number of 1 ÷ 1 = 1.0 — the best score any matrix can post.
Questions
What does the SVD-based condition number measure?
How unevenly a matrix stretches different directions, expressed as the ratio of its largest to smallest singular value, σ₁ ⁄ σ₂. A value close to 1 means the matrix stretches nearly every direction by the same amount; a large value means one direction gets amplified far more than another, which is exactly what makes a matrix numerically fragile to invert.
How is this different from this site's other Condition Number calculator?
That page computes κ using the infinity norm — the largest absolute row sum of the matrix and of its inverse, multiplied together. This page instead divides the largest singular value by the smallest, the 2-norm definition. Both are legitimately called 'the condition number,' but they are built from different rules and typically return different numbers for the same matrix; the singular-value version here is the more standard choice in numerical linear algebra.
How is this related to the Singular Values page on this site?
It reuses the identical σ₁ and σ₂ that page reports on their own, just dividing one by the other instead of listing both separately. If you already know a matrix's two singular values, you can get this page's answer yourself with a single division.
What is the smallest possible condition number, and which matrix achieves it?
1.0, the floor for any matrix, achieved whenever σ₁ equals σ₂ — the identity matrix is the simplest example, since it stretches every direction by exactly the same factor and therefore does not distort a circle into an ellipse at all.
Why does a large condition number matter in practice?
It flags a matrix as numerically fragile: solving a linear system or inverting a matrix with a large σ₁ ⁄ σ₂ can turn small measurement noise or rounding error into a large, unreliable swing in the answer, even when every arithmetic step was carried out correctly.
Can the condition number ever come out less than 1?
No — since σ₁ is defined as the larger singular value and σ₂ the smaller, their ratio can never fall below 1. A value of exactly 1 is the best case, reached only when a matrix stretches every direction identically.
What are the singular values of a diagonal matrix?
Simply the absolute values of its diagonal entries, listed largest first — for [[2,0],[0,3]], that means σ₁ = 3 and σ₂ = 2, no further calculation needed before dividing them.