2x2 Eigenvectors Calculator

Actual real2×2real-eigenvalue branch, sorted eigenvalues, unit vectors with first nonzero entry positive; repeated eigenvalue flag and one representative eigenspace vector, not claim diagonalizability or complex vectors.

Description

A real matrix can have real, repeated or complex eigenvalues, and repeated values do not ensure diagonalizability.

A real matrix can have real, repeated or complex eigenvalues, and repeated values do not ensure diagonalizability.^[1]

This tool computes the real-eigenvalue branch for a 2-by-2 matrix with normalized representative eigenvectors.

When to use 2x2 Eigenvectors Calculator

Use this tool to compute the real-eigenvalue branch for a 2-by-2 matrix with normalized representative eigenvectors.

How to use 2x2 Eigenvectors Calculator and how it works

Prepare actualMatrix, with the units, conventions and assumptions below.
Enter the required fields and run 2x2 Eigenvectors Calculator.
Read result.realEigenvalues, result.unitEigenvectors, result.repeatedEigenvalue 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 real matrix [[a,b],[c,d]]. Compute discriminant (a-d)^2+4bc and reject negative values, so complex eigenvalues are not returned. Sort the real eigenvalues ascending and find normalized real right eigenvectors using the larger off-diagonal entry, with the first nonzero vector component positive. A zero discriminant returns one eigenvalue and one selected vector with repeatedEigenvalue=true, even when the eigenspace has dimension two; this is not a complete repeated-eigenvalue basis or a diagonalizability test.

lambda=(a+d +/- sqrt((a-d)^2+4bc))/2; A*v=lambda*v; norm(v)=1

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.realEigenvalues	Real eigenvalues in ascending order; one value for a repeated root.	array of number
result.unitEigenvectors	Unit right eigenvectors corresponding to eigenvalue order; first nonzero component positive.	array of array of number
result.repeatedEigenvalue	True when the computed discriminant is exactly zero.	boolean

json

{
  "input": {
    "actualMatrix": [
      [
        2,
        0
      ],
      [
        0,
        1
      ]
    ]
  },
  "output": {
    "result": {
      "realEigenvalues": [
        1,
        2
      ],
      "unitEigenvectors": [
        [
          0,
          1
        ],
        [
          1,
          0
        ]
      ],
      "repeatedEigenvalue": false
    }
  }
}

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 Condition Number Calculator for a complementary operation: A spectral condition number compares the largest and smallest singular values of a matrix.

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

References

2x2 Eigenvectors Calculator — tool.io

Similar or alternative tools

2x2 Condition Number Calculator
A spectral condition number compares the largest and smallest singular values of a matrix.
2x2 Cofactor Calculator
A cofactor combines a deletion minor with a row-and-column sign.