2x2 Eigenvalues Calculator

Compute both algebraic eigenvalues of a real 2×2 matrix, preserving complex conjugate pairs and exact computed repeated-root flag; no false real-only truncation or eigenvector independence claim.

Description

A real 2×2 matrix can have real eigenvalues or a complex conjugate pair.

A real 2×2 matrix can have real eigenvalues or a complex conjugate pair. Repeated eigenvalues alone do not establish independent eigenvectors.^[1]

Compute both algebraic eigenvalues of a real 2×2 matrix, preserving complex conjugate pairs and exact computed repeated-root flag; no false real-only truncation or eigenvector independence claim.

When to use 2x2 Eigenvalues Calculator

Use this tool to compute both algebraic eigenvalues of a real 2×2 matrix, preserving complex conjugate pairs and exact computed repeated-root flag; no false real-only truncation or eigenvector independence claim.

Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.

How to use 2x2 Eigenvalues Calculator and understand its results

Choose the integer, sequence, coefficients or function parameters for your calculation. Follow the stated ordering, domain and size limits.
Enter matrix using the field table and calculation rules, then run the tool.
Read eigenvalueRealParts, eigenvalueImaginaryParts, repeatedEigenvalue. The explanation below defines the units, signs, ordering and any flags or placeholder values.
If the tool reports an input issue, correct the values and run it again. Apply the stated assumptions when interpreting a successfully returned result.

Input field	Meaning	Value type; see rules below	Requirement
matrix	Matrix	list of list of number	Required

Enter a finite 2×2 matrix. Normalize by its largest absolute entry, reject loss of nonzero entries during normalization and evaluate the discriminant. For real roots use a determinant-based second root to reduce cancellation; return aligned real/imaginary components and an exact computed repeated-root flag. The zero matrix returns two zero roots.

eigenvalues = (a+d)/2 ± sqrt(((a−d)/2)²+bc)

Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.

Output field	Meaning	Value type
eigenvalueRealParts	Eigenvalue real parts	list of number
eigenvalueImaginaryParts	Eigenvalue imaginary parts	list of number
repeatedEigenvalue	Repeated eigenvalue	boolean

json

{
  "input": {
    "matrix": [
      [
        0,
        -1
      ],
      [
        1,
        0
      ]
    ]
  },
  "output": {
    "eigenvalueRealParts": [
      0,
      0
    ],
    "eigenvalueImaginaryParts": [
      1,
      -1
    ],
    "repeatedEigenvalue": false
  }
}

Limitations and assumptions

Floating roundoff can affect an almost repeated root. No eigenvectors or diagonalizability verdict is returned.

Numeric inputs and returned numeric values must be finite; an unrepresentable returned value produces an error. Matrices are limited to 128 rows and 128 entries per row; other numeric lists are limited to 100,000 entries. Missing required fields, extra fields, wrong value types and violations of the calculation rules are input errors. A successfully returned false flag is an evaluated outcome; its meaning is explained above.

Alternative or Complementary methods

Use 2x2 Characteristic Polynomial Calculator: Construct det(λI−A) in ascending coefficient order for real 2×2 input, not a generic trace/determinant alias; preserves full characteristic polynomial. For a complementary task, use Vandermonde Determinant 3 Calculator: Compute oriented determinant of rows [1,x,x²] from three nodes without requiring caller prebuilt matrix; repeated nodes yield exact singular result.

References

2x2 Eigenvalues Calculator — tool.io

Similar or alternative tools

2x2 Characteristic Polynomial Calculator
Construct det(λI−A) in ascending coefficient order for real 2×2 input, not a generic trace/determinant alias; preserves full characteristic polynomial.
Vandermonde Determinant 3 Calculator
Compute oriented determinant of rows [1,x,x²] from three nodes without requiring caller prebuilt matrix; repeated nodes yield exact singular result.