General norm
κₚ(A) = ‖A‖ₚ ‖A⁻¹‖ₚThis calculator supports p = 1, 2, and ∞, plus the Frobenius norm.
Matrix Condition Number Calculator
Enter a Square Matrix
Enter integers, decimals, or scientific notation. Display precision never rounds intermediate calculations.
Condition and Stability
κ₂(A)
—
Enter a valid matrix
‖A‖₂—
‖A⁻¹‖₂—
Largest singular value σmax—
Smallest singular value σmin—
Estimated numerical rank—
Worst-case decimal digits lost—
The 2-norm condition number is the ratio σmax/σmin. Calculate a matrix to see its sensitivity.
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Inverse Matrix
Shown as a useful verification output; the inverse is not formed to calculate κ₂.
Calculate a nonsingular matrix to display A⁻¹.
Condition Number Formulas
For an invertible square matrix, a normwise condition number multiplies the size of the matrix by the size of its inverse. The value is scale-invariant: multiplying every entry by the same nonzero constant does not change it.
Spectral 2-norm
κ₂(A) = σmax(A) / σmin(A)The singular-value ratio is the standard general-purpose measure for sensitivity in Euclidean distance.
Singular matrix
det(A) = 0 ⇒ κ(A) = ∞A singular matrix has no inverse. A nearly singular matrix has a very small σmin and a very large finite condition number.
Numerical method: The calculator rescales the matrix before computing, uses one-sided Jacobi rotations for singular values, and uses pivoted elimination for the displayed inverse and non-spectral norms. All arithmetic is IEEE 754 double precision.
How to Use the Calculator
- Choose the square matrix size and the norm that matches your analysis. The 2-norm is a sensible default.
- Enter every matrix entry, or paste complete rows from a spreadsheet or text file.
- Select Calculate condition number. Review κ, σmin, σmax, and the estimated numerical rank.
- Use the digit-loss estimate as a worst-case sensitivity guide, then copy the summary or download the results as CSV.
How to Interpret a Condition Number
A condition number describes possible error amplification, not the error of one particular answer. Roughly, an input perturbation of relative size ε can produce a relative output change on the order of κ(A)ε in a matching norm.
| κ magnitude | Rough reading | Potential decimal digits lost |
|---|---|---|
| Near 1 | Low sensitivity | Near 0 |
| 10² | Noticeable amplification | Up to about 2 |
| 10⁸ | Strongly ill-conditioned | Up to about 8 |
| ∞ | Singular; inverse does not exist | No finite bound |
These descriptions are orientation only. Acceptable conditioning depends on data accuracy, algorithm, precision, scaling, and the consequences of error. The Frobenius condition number has a baseline of n for an n×n identity matrix, rather than 1.
Examples: Stable, Ill-Conditioned, and Singular
Identity matrix
A = [[1,0],[0,1]]κ₂(A) = 1. The matrix preserves Euclidean distances and is optimally conditioned in the 2-norm.
Nearly dependent rows
A = [[1,1],[1,1.000001]]The rows are almost identical, σmin is tiny, and κ₂ is large. Small entry changes can strongly affect solutions.
Exactly dependent rows
A = [[1,2],[2,4]]The second row is twice the first. The matrix is singular, so its condition number is infinite.
Matrix Condition Number FAQs
What does a matrix condition number measure?
It measures worst-case relative sensitivity to small input changes under a chosen norm. A high value warns that rounding or measurement error may be amplified.
Which norm should I choose?
Use the 2-norm for a common Euclidean sensitivity measure. The 1-norm uses the largest absolute column sum, while the infinity norm uses the largest absolute row sum. The Frobenius norm treats all entries together but is not an induced operator norm.
What condition number is considered bad?
There is no universal cutoff. Compare κ(A) with the precision of your inputs and arithmetic. In double precision, a condition number near 10¹⁶ can consume essentially all available decimal precision in a worst case.
Why is a singular matrix’s condition number infinite?
A singular matrix has no inverse and maps at least one nonzero direction to zero. Arbitrarily small perturbations can therefore change solvability or the solution dramatically.
Does a small residual prove a solution is accurate?
No. An ill-conditioned system can have a small residual while its computed solution is far from the exact solution. Conditioning links backward error to possible forward error.
Will scaling the entire matrix change κ?
No. Multiplying every entry by one nonzero scalar changes the matrix norm and inverse norm by reciprocal factors. Scaling individual rows or columns can change conditioning.
Are my matrix values private?
Yes. Parsing, calculation, clipboard text, and CSV creation happen locally in your browser. This calculator’s code does not transmit or save matrix entries.
Calculation Notes and Limits
Last reviewed: August 4, 2026 by the Starlight Tools editorial team.
Inputs must be finite and no larger than 10¹⁰⁰ in magnitude. Matrices are limited to 8×8 for readable interaction. Estimated rank uses a double-precision threshold based on matrix size and σmax; a matrix near that threshold may be classified differently by higher-precision or specialist numerical software. This educational tool does not replace validated numerical analysis for safety-critical engineering or scientific decisions.