Mathematics calculator

Eigenvalue Calculator

Find both eigenvalues of a 2 × 2 matrix, including complex conjugate values when the characteristic discriminant is negative.

Calculator inputs

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2 × 2 matrix
Calculated output

Result summary

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Complete the required fields and select the calculate button.

Table of Contents

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What is Eigenvalue?

An eigenvalue describes how a square matrix stretches, shrinks or reverses a special direction called an eigenvector. For a 2 by 2 matrix, the calculator finds both eigenvalues from the four entries. This is useful when checking a linear-algebra exercise, studying repeated transformations, or examining a small system whose long-term behavior depends on its eigenvalues.

The result can contain two different real values, one repeated real value, or a complex-conjugate pair. Eigenvalues are properties of the whole matrix, so a result should not be read as a separate calculation for each row. The calculator accepts decimal and negative entries and reports up to four decimal places.

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How to Use This Calculator

  1. Arrange the matrix as a1 and a2 on the first row, with b1 and b2 on the second row. Copy every sign carefully; changing one off-diagonal sign can change the determinant and both answers.
  2. Enter all four entries. The values have no units unless the matrix in your problem has a defined physical interpretation.
  3. Select Calculate. The trace is found by adding the main diagonal, and the determinant is found from the difference of the two diagonal products.
  4. The two roots of the characteristic equation are displayed. A letter i means the matrix has a complex pair rather than real eigenvalues.
  5. For a check, add the two displayed eigenvalues and compare the sum with the trace. Their product should equal the determinant, allowing for display rounding.
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Formula Used

For A = [[a1, a2], [b1, b2]], the trace is T = a1 + b2 and the determinant is Δ = a1b2 − a2b1. The characteristic equation is λ² − Tλ + Δ = 0.

The solutions are λ = (T ± √(T² − 4Δ)) / 2. If T² − 4Δ is negative, write √(−q) as i√q. The real part of each eigenvalue is T/2 and the imaginary parts have equal magnitude with opposite signs.

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Variables Explained

InputMeaning and use
a1Top-left matrix entry. It contributes to both the trace and determinant.
a2Top-right matrix entry. Together with b1 it forms the off-diagonal product.
b1Bottom-left matrix entry. Its sign matters in the term a2b1.
b2Bottom-right matrix entry. It contributes to both the trace and determinant.
λ1 and λ2The two roots of the characteristic equation. Their order has no mathematical significance.
iThe imaginary unit, defined by i² = −1; it appears when the discriminant is negative.
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Worked Example

Take A = [[4, 2], [1, 3]]. Its trace is 4 + 3 = 7. Its determinant is (4 × 3) − (2 × 1) = 10. The characteristic equation is therefore λ² − 7λ + 10 = 0.

Factoring gives (λ − 2)(λ − 5) = 0, so the calculator reports λ1 = 2 and λ2 = 5. The checks agree: 2 + 5 equals the trace 7, while 2 × 5 equals the determinant 10. For a complex example, [[0, −1], [1, 0]] has trace 0, determinant 1 and discriminant −4, producing −1i and +1i.

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Why Use This Calculator?

An eigenvalue gives the multiplier along its matching eigenvector. A value larger than 1 in absolute value represents growth under repeated application in that direction; a magnitude between 0 and 1 represents decay. A negative real value also reverses direction each time. Zero means the matrix collapses at least one nonzero direction to zero and is not invertible.

Complex values often describe rotation combined with scaling. Their magnitude, rather than either printed component by itself, governs scaling. Context still matters: stability rules differ between discrete-time and continuous-time models, and eigenvalues alone do not supply eigenvectors or establish every property of a matrix. A complete analysis may also require eigenvectors, multiplicities and the model’s time convention. Keep the original matrix available when interpreting either root.

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Common Mistakes

  • Entering the matrix by columns instead of rows changes the matrix unless the intended operation is a transpose.
  • Using a1b1 − a2b2 for the determinant. The correct 2 by 2 determinant is a1b2 − a2b1.
  • Dropping a negative sign in an off-diagonal entry, especially when subtracting the off-diagonal product.
  • Assuming a negative discriminant means there is no answer. It means the answers are a complex-conjugate pair.
  • Treating rounded eigenvalues as exact when checking their sum or product. Allow for the four-decimal display.
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Frequently Asked Questions

Does the order of λ1 and λ2 matter?

No. The two eigenvalues form an unordered set. A textbook or another calculator may list the larger value first without disagreeing with this result.

Why did I receive an answer containing i?

The characteristic discriminant is negative, so no real number can satisfy the equation. A real 2 by 2 matrix can still have two complex-conjugate eigenvalues.

Can both eigenvalues be the same?

Yes. When the discriminant is zero, both roots equal half the trace. A repeated eigenvalue does not by itself say whether the matrix has one or two independent eigenvectors.

What is a quick way to check the result?

Add the eigenvalues to recover the trace and multiply them to recover the determinant. With displayed decimals, the checks may differ slightly because of rounding.

Does this calculator find eigenvectors too?

No. It finds only the two eigenvalues. Finding an eigenvector requires solving (A − λI)v = 0 separately for each eigenvalue.

Can I use this for a 3 by 3 matrix?

No. The four fields represent a 2 by 2 matrix only. A 3 by 3 problem has a cubic characteristic polynomial and needs nine entries.

What does a zero eigenvalue tell me?

It tells you the determinant is zero, so the matrix is singular. At least one nonzero direction is mapped to the zero vector.

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Sources & References