2x2 Condition Number Calculator

Actual2×2spectral2norm condition numberσmax/σmin, zero/singular cases rejected rather than fabricate finite value; scaled stable singular values, not1norm/∞norm.

Description

A spectral condition number compares the largest and smallest singular values of a matrix.

A spectral condition number compares the largest and smallest singular values of a matrix.^[1]

This tool computes the 2-norm ratio for a nonsingular 2-by-2 matrix rather than a different matrix norm.

When to use 2x2 Condition Number Calculator

Use this tool to compute the 2-norm ratio for a nonsingular 2-by-2 matrix rather than a different matrix norm.

How to use 2x2 Condition Number Calculator and how it works

Prepare actualMatrix, with the units, conventions and assumptions below.
Enter the required fields and run 2x2 Condition Number Calculator.
Read result.spectralConditionNumber using the output definitions below.
Correct any rejected inputs and rerun; interpret the result within the stated model and its assumptions.

Input field	Meaning	Value type	Requirement
actualMatrix	Exactly two rows of two finite entries, using the row and column ordering below.	array of array of number	Required; other constraints below

Require a finite 2-by-2 matrix. Scale its entries by their largest absolute value, derive its two singular values from the trace and determinant of the normalized Gram matrix, and return largest/smallest. Reject a zero smallest singular value rather than returning infinity; near-singular matrices remain subject to binary64 numerical error.

conditionNumber=sigma_max(A)/sigma_min(A)

Apply the input ordering, units and branch conditions stated above.

Output field	Meaning	Value type
result	Object containing the calculated or selected fields below.	object
result.spectralConditionNumber	Largest singular value divided by the positive smallest singular value; dimensionless.	number

json

{
  "input": {
    "actualMatrix": [
      [
        2,
        0
      ],
      [
        0,
        1
      ]
    ]
  },
  "output": {
    "result": {
      "spectralConditionNumber": 2
    }
  }
}

Limitations and assumptions

The calculation uses the supplied observations and model inputs. It does not collect measurements, select a reference population or coefficient, or establish that the model applies to the actual case. The stated branch conditions and output units remain part of the result.

All fields require explicit values. Unexpected fields, wrong types, nonfinite numbers and sparse arrays are rejected. Alignment, bounds and supported-token rules are explained above; extremely large inputs or nonfinite results can also fail. Input errors differ from successful false comparisons, unavailable-domain flags and empty match lists.

Alternative or Complementary methods

Use 2x2 Eigenvectors Calculator for a complementary operation: A real matrix can have real, repeated or complex eigenvalues, and repeated values do not ensure diagonalizability.

Use 2x2 Cofactor Calculator for a complementary operation: A cofactor combines a deletion minor with a row-and-column sign.

References

2x2 Condition Number Calculator — tool.io

Similar or alternative tools

2x2 Eigenvectors Calculator
A real matrix can have real, repeated or complex eigenvalues, and repeated values do not ensure diagonalizability.
2x2 Cofactor Calculator
A cofactor combines a deletion minor with a row-and-column sign.