Softonic

Eigenvalue Calculator

Step 1 / 4 Matrix A
First row

Use this eigenvalue calculator to solve a real 2×2 matrix from its four entries. The tool computes the trace, determinant, characteristic polynomial, discriminant and eigenvalues, then shows real eigenvectors when the two eigenvalues are distinct and real. It is designed for linear algebra homework, quick checks in engineering models and sanity checks before diagonalizing a small matrix by hand.

How to find the eigenvalues

  1. 1

    Enter the matrix entries

    Fill in a, b, c and d for the matrix A = [[a, b], [c, d]]. Decimals and negative values are accepted.

  2. 2

    Build the characteristic equation

    The calculator uses trace T = a + d and determinant D = ad - bc to form λ² - Tλ + D = 0.

  3. 3

    Classify the roots

    The discriminant T² - 4D determines whether the eigenvalues are two real values, one repeated value or a complex conjugate pair.

Formula for a 2×2 matrix

For A = [[a, b], [c, d]], the eigenvalues are the roots of:

det(A - λI) = 0

Expanding that determinant gives:

λ² - Tλ + D = 0

Where:

  • T = a + d is the trace.
  • D = ad - bc is the determinant.
  • Δ = T² - 4D is the discriminant.

Then:

λ = (T ± sqrt(Δ)) / 2

Worked example

For A = [[2, 1], [1, 2]], the trace is T = 2 + 2 = 4 and the determinant is D = 2·2 - 1·1 = 3. The characteristic polynomial is:

λ² - 4λ + 3 = 0

The discriminant is Δ = 4² - 4·3 = 4, so the eigenvalues are:

λ₁ = (4 + 2) / 2 = 3

λ₂ = (4 - 2) / 2 = 1

For the eigenvalue 3, one eigenvector is [1, 1]. For the eigenvalue 1, one eigenvector is [1, -1]. Any non-zero scalar multiple of those vectors is also a valid eigenvector.

What the discriminant means

Discriminant Δ Eigenvalue case What you should expect
Δ > 0 Two real eigenvalues Two distinct real roots and, for a 2×2 matrix, two independent eigenvectors when the matrix is diagonalizable over the reals.
Δ = 0 Repeated eigenvalue One repeated root. The eigenspace may have one or two dimensions, so check eigenvectors separately if diagonalization matters.
Δ < 0 Complex conjugate pair No real eigenvalues. The roots have the same real part and opposite imaginary parts.

Common mistakes

  • Using A - λI incorrectly. Only the diagonal entries change: a - λ and d - λ.
  • Forgetting the determinant sign. For a 2×2 matrix, D = ad - bc, not ad + bc.
  • Treating repeated eigenvalues as automatically diagonalizable. A repeated root still needs enough independent eigenvectors.
  • Rounding too early. Keep the trace, determinant and discriminant exact as long as possible, especially with decimals.

Frequently Asked Questions

This tool focuses on real 2×2 matrices. That keeps the result transparent: every value comes from trace, determinant and the quadratic characteristic polynomial.

Yes. If the discriminant T² - 4D is negative, the eigenvalues form a complex conjugate pair. A rotation matrix such as [[0, -1], [1, 0]] is a standard example.

The calculator shows eigenvectors for distinct real eigenvalues, where a simple real vector can be displayed for each root. Repeated and complex cases need extra context, so the tool focuses on the eigenvalues and classification there.

No file upload is involved. The entries are evaluated by the page component to produce the trace, determinant, polynomial and eigenvalues.

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