Condition Number Calculator

The condition number κ(A) measures how sensitive the solution of a linear system Ax = b is to small changes or errors in A or b. It is defined as κ(A) = ‖A‖ · ‖A⁻¹‖ when A is invertible, and ∞ otherwise. A condition number close to 1 means the matrix is well-conditioned; a very large condition number signals that the matrix is ill-conditioned and results may be numerically unreliable.

To find the condition number, compute the chosen matrix norm of A (e.g., the 2-norm), then compute the same norm of its inverse A⁻¹, and multiply them together: κ(A) = ‖A‖ · ‖A⁻¹‖. For the 2-norm specifically, this equals the ratio of the largest to the smallest singular value of A.

A large condition number (e.g., greater than 10³ or 10⁶) indicates an ill-conditioned matrix, meaning tiny perturbations in the input data can cause large changes in the solution. This is particularly important in numerical linear algebra, where floating-point rounding can amplify errors dramatically in ill-conditioned systems.

The condition number of the identity matrix I is always 1, regardless of the norm used. This is the best possible condition number, meaning the identity matrix is perfectly well-conditioned — any linear system with I as its coefficient matrix is completely stable.

For a diagonal matrix, the condition number (using the 2-norm) is the ratio of the largest diagonal entry (in absolute value) to the smallest. If any diagonal entry is zero, the matrix is singular and the condition number is infinite.

No, the condition number cannot be zero. Since ‖A‖ ≥ 0 and ‖A⁻¹‖ ≥ 0, and a valid invertible matrix always has a positive norm, the minimum possible condition number is 1 (achieved by scalar multiples of unitary or orthogonal matrices). A condition number of zero would imply a zero norm, which is impossible for a non-zero matrix.

Scaling a matrix by a scalar constant c does not change its condition number, because both ‖cA‖ and ‖(cA)⁻¹‖ = (1/c)‖A⁻¹‖ scale in opposite directions, canceling out. However, row or column scaling by different factors can change the condition number significantly, which is the basis of matrix preconditioning techniques.

This calculator supports the 1-norm (maximum absolute column sum), 2-norm (largest singular value / spectral norm), ∞-norm (maximum absolute row sum), and Frobenius norm (square root of sum of squared entries). The 2-norm is the most commonly used in condition number analysis and is the default in tools like MATLAB's cond() function. The choice of norm affects the numerical value of κ(A), but all norms are equivalent up to a constant factor.