Eigenvalue Calculator

Eigenvalue is evaluated from A[1,1], A[1,2] and A[2,1]. The calculation reports Trace, Determinant and Discriminant.

Results

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About the Eigenvalue Calculator

Eigenvalue is treated here as a quantitative relation between A[1,1], A[1,2], A[2,1] and A[2,2] and Trace, Determinant, Discriminant and Eigenvalue λ₁.

The calculator uses a multi formula configuration. Each reported value is read as a direct evaluation of the stored rules with the declared field formats and units.

Formula basis:
Eigenvalues λ satisfy Av = λv where v ≠ 0. Setting det(A - λI) = 0 gives characteristic polynomial. For 2 x 2: quadratic with coefficients from trace and determinant. Discriminant sign determines real vs complex eigenvalues.

Interpret the outputs in the order shown by the result fields. Optional inputs affect only the outputs that depend on those variables.

Formula & How It Works

The calculation applies the following relations exactly as recorded in the metadata:

Eigenvalues λ satisfy Av = λv where v ≠ 0. Setting det(A - λI) = 0 gives characteristic polynomial. For 2 x 2: quadratic with coefficients from trace and determinant. Discriminant sign determines real vs complex eigenvalues.

Each output field is produced by substituting the supplied inputs into the relevant relation and then applying the declared rounding or text format.

Worked Examples

Example 1: Standard matrix [[4,1],[2,3]]

Inputs

a11: 4 a12: 1 a21: 2 a22: 3

Trace: 7. Determinant: 10. Discriminant: 9. Eigenvalue λ₁: 5. Eigenvalue λ₂: 2. Eigenvalue Type: Two distinct real eigenvalues

With A[1,1] = 4, A[1,2] = 1, A[2,1] = 2 and A[2,2] = 3 as the stated inputs, the result is Trace = 7, Determinant = 10 and Discriminant = 9. Each value corresponds to the declared output fields.

Example 2: Symmetric matrix [[2,1],[1,2]]

Inputs

a11: 2 a12: 1 a21: 1 a22: 2

Trace: 4. Determinant: 3. Discriminant: 4. Eigenvalue λ₁: 3. Eigenvalue λ₂: 1. Eigenvalue Type: Two distinct real eigenvalues

With A[1,1] = 2, A[1,2] = 1, A[2,1] = 1 and A[2,2] = 2 as the stated inputs, the result is Trace = 4, Determinant = 3 and Discriminant = 4. Each value corresponds to the declared output fields.

Example 3: Repeated eigenvalue [[3,0],[0,3]]

Inputs

a11: 3 a12: 0 a21: 0 a22: 3

Trace: 6. Determinant: 9. Discriminant: 0. Eigenvalue λ₁: 3. Eigenvalue λ₂: 3. Eigenvalue Type: Repeated real eigenvalue

With A[1,1] = 3, A[1,2] = 0, A[2,1] = 0 and A[2,2] = 3 as the stated inputs, the result is Trace = 6, Determinant = 9 and Discriminant = 0. Each value corresponds to the declared output fields.

Example 4: Rotation-like matrix [[1,-2],[2,1]]

Inputs

a11: 1 a12: -2 a21: 2 a22: 1

Trace: 2. Determinant: 5. Discriminant: -16. Eigenvalue λ₁: 1. Eigenvalue λ₂: 1. Eigenvalue Type: Complex conjugate eigenvalues (no real eigenvalues)

With A[1,1] = 1, A[1,2] = -2, A[2,1] = 2 and A[2,2] = 1 as the stated inputs, the result is Trace = 2, Determinant = 5 and Discriminant = -16. Each value corresponds to the declared output fields.

Common Use Cases

Calculate eigenvalues for 2×2 matrices in linear algebra
Determine if matrix is positive definite (all positive eigenvalues)
Analyze stability of linear dynamical systems