Calculator · Engineering Mathematics
Matrix Condition Number Calculator
Measure how sensitive a linear problem with this matrix is to small changes in its data. Conditioning is not the same thing as residual: a computed solution can satisfy the equations almost exactly and still be wide open to perturbation.
Matrix A
One row per line; entries separated by spaces or commas. Scientific notation such as 1.5e-3 is accepted. At most 10 × 10; ragged rows and blank entries are refused, never repaired.
What Anemodia read
| 1 | 0.5 | 0.3333 |
| 0.5 | 0.3333 | 0.25 |
| 0.3333 | 0.25 | 0.2 |
Results
| Size | 3 × 3 |
|---|---|
| Numerical rank | 3 of 3 |
| Rank tolerance (eps · n · σmax) | 9.381e-16 |
| σ max | 1.408 |
| σ min | 0.002687 |
| κ₂ = σmax / σmin | 524.1 |
| κ₁ = ‖A‖₁‖A⁻¹‖₁ | 748 |
| κ∞ = ‖A‖∞‖A⁻¹‖∞ | 748 |
| rcond = 1/κ₂ | 0.001908 |
| log₁₀ κ₂ (approximate digits indicator) | 2.719 |
| ‖A‖₁ (max column sum) | 1.833 |
| ‖A‖∞ (max row sum) | 1.833 |
| ‖A‖₂ = σmax (induced 2-norm) | 1.408 |
| ‖A‖_F (Frobenius, a different quantity) | 1.414 |
| 1.408 |
| 0.1223 |
| 0.002687 |
Continue
The matrix continues into the Lab's Condition workspace, where you can also run an explicit perturbation of b and compare the observed amplification with κ.
Assumptions & validity
- · A must be square and at most 10 × 10.
- · κ₂ comes from the singular values. κ₁ and κ∞ use an inverse obtained through stable triangular solves; A is never inverted in order to solve a system elsewhere.
- · A singular matrix has κ = ∞ exactly and rcond = 0. No large finite number is invented by dividing by a rounding-level singular value.
- · A singular value counts as zero below eps · n · σmax. That threshold scales with the matrix, so κ(cA) = κ(A) for every finite non-zero c.
- · κ is a sensitivity measure, not an error. It bounds how much a relative perturbation CAN be amplified; a particular perturbation may be amplified far less.
Technical reference
κ_p(A) = ‖A‖_p ‖A⁻¹‖_p
κ₂(A) = σ_max / σ_min
rcond = 1 / κ₂, singular → κ = ∞, rcond = 0
‖δx‖/‖x‖ ≲ κ(A) · (relative perturbation of the data)
- σ
- singular value of A, the semi-axis lengths of the image of the unit sphere
- κ₂
- condition number in the induced 2-norm
- rcond
- reciprocal condition number, 0 for a singular matrix
- ‖A‖₁ / ‖A‖∞
- maximum absolute column sum / row sum
For any induced norm κ(A) ≥ 1, because ‖A‖‖A⁻¹‖ ≥ ‖A A⁻¹‖ = 1. The identity attains that minimum: κ(I) = 1.
κ is invariant under scaling: multiplying A by any finite non-zero c multiplies both ‖A‖ and ‖A⁻¹‖⁻¹ by |c|. Conditioning is a property of the geometry of the map, not of the units it is written in.
The relation between perturbation and solution change is a BOUND under the appropriate assumptions, not an equality. Writing 'relative output error = κ × relative input error' would overstate it: the observed amplification depends on the direction of the perturbation and is usually much smaller.
log₁₀κ₂ is sometimes read as the number of decimal digits that may be lost. Treat it as an approximate indicator only — it assumes a worst-case perturbation and says nothing about the particular data you have.
The residual r = b − Ax says how well the computed vector satisfies the equations. It is NOT the forward error x_computed − x_true, which cannot be known without the exact solution. A tiny residual and a highly sensitive problem regularly occur together.
Rules of thumb such as 'κ > 10⁸ is bad' depend entirely on the precision you are working in and on how accurate your data is in the first place. They are rules of thumb, and this page does not turn one into a verdict.