Condition Number — Definition, Formula & Examples

The condition number of a matrix measures how sensitive its output is to small changes in input. A large condition number means even tiny errors in data can produce large errors in the solution, making the system ill-conditioned.

For a nonsingular matrix $A$ and a given matrix norm $\|\cdot\|$ , the condition number is defined as $\kappa(A) = \|A\|\,\|A^{-1}\|$ . It satisfies $\kappa(A) \geq 1$ , with $\kappa(A) = 1$ for orthogonal (or unitary) matrices. When $\kappa(A)$ is large, the linear system $Ax = b$ is ill-conditioned, meaning the relative error in $x$ can be amplified by a factor of $\kappa(A)$ relative to the perturbation in $b$ or $A$ .

Key Formula

\kappa(A) = \|A\|\,\|A^{-1}\|

Where:

$A$ = A nonsingular (invertible) square matrix
$\|\cdot\|$ = A consistent matrix norm (e.g., 2-norm, Frobenius norm)
$\kappa(A)$ = The condition number of A with respect to the chosen norm

How It Works

When you solve

Ax = b

numerically, rounding errors and measurement noise perturb

b

by some small relative amount

\frac{\|\delta b\|}{\|b\|}

. The condition number bounds the worst-case amplification:

\frac{\|\delta x\|}{\|x\|} \leq \kappa(A)\,\frac{\|\delta b\|}{\|b\|}

. If

\kappa(A) = 10^k

, you can lose roughly

k

digits of accuracy in your solution. A system with

\kappa(A)

near 1 is called well-conditioned; one with

\kappa(A) \gg 1

is ill-conditioned. Checking the condition number before solving a linear system is standard practice in scientific computing.

Worked Example

Problem: Find the 2-norm condition number of the matrix

A = \begin{pmatrix} 1 & 0 \\ 0 & 0.001 \end{pmatrix}

Step 1: For the 2-norm, the condition number equals the ratio of the largest singular value to the smallest. Since

A

is diagonal, its singular values are the absolute values of the diagonal entries:

\sigma_1 = 1

and

\sigma_2 = 0.001

\sigma_{\max} = 1, \quad \sigma_{\min} = 0.001

Step 2: Compute the condition number as the ratio of these singular values.

\kappa_2(A) = \frac{\sigma_{\max}}{\sigma_{\min}} = \frac{1}{0.001} = 1000

Answer:

\kappa_2(A) = 1000

. This means perturbations in

b

can be amplified by up to a factor of 1000 in the solution

x

, so roughly 3 digits of accuracy could be lost.

Why It Matters

In engineering simulations, weather modeling, and machine learning, you routinely solve large linear systems. Checking the condition number tells you whether your computed answer can be trusted or whether you need techniques like preconditioning, regularization, or higher-precision arithmetic to obtain reliable results.

Common Mistakes

Mistake: Assuming a large determinant means the matrix is well-conditioned.

Correction: Determinant magnitude and condition number are unrelated. A matrix can have a large determinant yet be extremely ill-conditioned (e.g., a large diagonal matrix with entries of vastly different magnitudes). Always compute

\kappa(A)

directly.