Calculate the condition number of a matrix
Enter rows on separate lines and values separated by commas
Last updated: 2026-05-24T22:58:31.724Z | By ForgeCalc Engineering
Example: 1,2,3 4,5,6
Enter rows separated by newlines and columns separated by commas. All rows must have the same number of columns and contain numeric values.
Condition Number
14.933034
σ Max
5.464986
σ Min
0.365966
Interpretation
Moderately conditioned
The condition number of a matrix measures how sensitive the solution of a linear system is to small changes in the input data. In practical terms, it tells you whether rounding errors, measurement noise, or tiny perturbations can cause large changes in the answer.
A condition number close to 1 indicates a well-conditioned matrix, which means computations are stable. A large condition number indicates an ill-conditioned matrix, meaning the system may be numerically unstable and small errors can be greatly amplified.
This calculator estimates the matrix condition number using the 2-norm, which is based on the ratio of the largest singular value to the smallest singular value.
For the matrix 2-norm, the condition number is:
κ(A) = σmax(A) ÷ σmin(A)
Here, σmax is the largest singular value of the matrix and σmin is the smallest singular value. If the smallest singular value is very close to zero, the condition number becomes very large, indicating that the matrix is nearly singular.
Step 1: Enter the Matrix
Type each row on a new line and separate entries in a row with commas. For example, a 2×2 matrix can be entered as:1,23,4
Step 2: Form ATA
The calculator multiplies the transpose of the matrix by the matrix itself. This creates a symmetric matrix whose eigenvalues are related to the singular values of the original matrix.
Step 3: Estimate Singular Values
The largest and smallest singular values are estimated numerically. These are reported as σ max and σ min in the results panel.
Step 4: Divide Largest by Smallest
The condition number is found by dividing the largest singular value by the smallest singular value. A larger result indicates greater sensitivity to numerical error.
Consider the matrix:
[ 1 2 ]
[ 3 4 ]
Step 1: Enter the matrix as:
1,2
3,4
Step 2: The calculator estimates the singular values.
Step 3: It divides the largest singular value by the smallest singular value.
Result: The matrix has a condition number greater than 1, so it is not perfectly conditioned. The larger the value, the more numerical sensitivity the matrix has.
Near 1
The matrix is well-conditioned. Numerical computations are generally stable and resistant to small perturbations.
10 to 100+
The matrix has moderate to noticeable sensitivity. Errors may begin to affect computed solutions.
Very Large
The matrix is ill-conditioned or nearly singular. Even tiny input errors can produce large output errors.
What does a high condition number mean?
A high condition number means the matrix is sensitive to small changes. Tiny rounding or input errors can produce much larger changes in the computed solution.
What is a good condition number?
A condition number close to 1 is usually considered very good. It indicates a well-conditioned matrix with stable numerical behavior.
Can the condition number be less than 1?
No. For the matrix 2-norm, the condition number is always at least 1 for nonsingular matrices because the largest singular value cannot be smaller than the smallest singular value.
What happens if the matrix is singular?
A singular matrix has a smallest singular value of zero, so its condition number is effectively infinite. That indicates the matrix cannot be inverted.
Does the calculator support rectangular matrices?
Yes. The calculator accepts rectangular matrices and computes the condition number using singular-value-based logic from AᵀA.
Why use singular values instead of determinants?
Determinants do not reliably measure numerical sensitivity. Singular values directly reflect how the matrix stretches or compresses vectors, which is why they are used for condition numbers.
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