Matrix Eigenvalues Calculator – Find Eigenvalues Online

Calculate matrix eigenvalues

Select a matrix size, enter its values, and calculate the eigenvalues. This calculator supports real-valued matrices of size 1×1 and 2×2.

Square matrix size

Enter finite real numbers between −1,000,000 and 1,000,000.

What are matrix eigenvalues?

An eigenvalue is a number that describes how a square matrix acts on a particular nonzero vector. If multiplying a matrix by a vector produces a scalar multiple of that vector, the scalar is an eigenvalue.

Av = λv

Here, A is a square matrix, v is a nonzero eigenvector, and λ is its corresponding eigenvalue. Rearranging gives (A − λI)v = 0. A nonzero solution exists only when the matrix A − λI is singular.

det(A − λI) = 0

This equation is called the characteristic equation. Its roots are the eigenvalues of A.

Formula for 2×2 matrix eigenvalues

For the matrix

A = [ a b ]
[ c d ]

The characteristic equation is

λ² − (a + d)λ + (ad − bc) = 0

The trace is a + d and the determinant is ad − bc. Applying the quadratic formula gives:

λ = (tr(A) ± √(tr(A)² − 4 det(A))) / 2

Understanding the discriminant

Positive

Δ > 0: two distinct real eigenvalues.

Zero

Δ = 0: a repeated real eigenvalue.

Negative

Δ < 0: a complex-conjugate pair.

Worked example: find two eigenvalues

Consider the matrix

A = [ 4 1 ]
[ 2 3 ]

Step 1: Find the trace and determinant

tr(A) = 4 + 3 = 7
det(A) = (4 × 3) − (1 × 2) = 10

Step 2: Form the characteristic equation

λ² − 7λ + 10 = 0

Step 3: Factor the polynomial

(λ − 5)(λ − 2) = 0

Step 4: Read the eigenvalues

λ₁ = 5, λ₂ = 2

The eigenvalues sum to 7, which is the trace, and multiply to 10, which is the determinant. Select the 2×2 option and load the example above to check the result.

Example with complex eigenvalues

A real matrix can have eigenvalues that are complex numbers. Consider a rotation matrix:

A = [ 0 −1 ]
[ 1 0 ]

Its trace is 0 and its determinant is 1. Therefore, its characteristic equation is:

λ² + 1 = 0

λ₁ = i, λ₂ = −i

These eigenvalues are a complex-conjugate pair. The calculator displays the real and imaginary parts of such eigenvalues.

Properties of eigenvalues

Sum of eigenvalues

For a 2×2 matrix, the eigenvalues counted with multiplicity add up to the trace of the matrix.

λ₁ + λ₂ = tr(A)

Product of eigenvalues

The product of the eigenvalues, counted with multiplicity, equals the determinant.

λ₁ × λ₂ = det(A)

Triangular matrices

For an upper- or lower-triangular matrix, the eigenvalues are its diagonal entries.

Frequently asked questions

What is an eigenvalue of a matrix?

An eigenvalue is a scalar λ for which a square matrix A has a nonzero vector x satisfying Ax = λx. The matrix transforms that vector by scaling it by λ.

How do you calculate eigenvalues of a 2×2 matrix?

Find the trace and determinant, then solve λ² − tr(A)λ + det(A) = 0 using the quadratic formula.

Can eigenvalues be complex numbers?

Yes. A real matrix can have complex eigenvalues. For a real 2×2 matrix, a negative characteristic-equation discriminant produces a complex-conjugate pair.

What does a repeated eigenvalue mean?

A repeated eigenvalue appears more than once as a root of the characteristic polynomial. A repeated eigenvalue does not necessarily mean there are multiple independent eigenvectors.

What is the characteristic equation?

The characteristic equation is det(A − λI) = 0. Its roots are the eigenvalues of the square matrix A.

Can this calculator solve a 3×3 or larger matrix?

This version supports 1×1 and 2×2 matrices. General eigenvalue calculations for larger matrices require more advanced numerical methods, so those sizes are not offered by this calculator.

What is the relationship between eigenvalues, trace, and determinant?

For a 2×2 matrix, the sum of its two eigenvalues equals the trace, and their product equals the determinant, counting repeated eigenvalues with multiplicity.

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