Eigenvalue and Eigenvector Calculator

Find the characteristic polynomial, eigenvalues, and eigenspaces of a square matrix.

Drag & drop or upload an image or PDF

Enter a square matrix, such as eigenvalues([[4,1],[2,3]])

Enter rows in one line, such as [[1, 2], [3, 4]]. State the operation when the page accepts more than one.

Try an example

Eigenvectors keep their direction under a matrix

An eigenvector is a nonzero vector whose direction is preserved by the matrix transformation, apart from a possible reversal. Its eigenvalue is the scalar factor in Av = λv.

Eigenvalues come from the roots of det(A - λI). For each root, solve the null space of A - λI to find the eigenspace. Repeated roots need extra care because algebraic multiplicity does not guarantee the same number of independent eigenvectors.

How to use the eigenvalue calculator

Enter the problem as written

Type or paste the full expression. You can also upload a clear photo or PDF and check the extracted text before solving.

Read the working, not only the answer

Each transformation is separated and explained so you can compare it with your own method.

Ask about any step

Continue in the same solution to request another method, check a restriction, or ask why a rule applies.

The characteristic equation

Values of λ that make A - λI singular are the eigenvalues. Their nonzero null-space vectors are the corresponding eigenvectors.

det (A - λ I) = 0

A reliable way to work through it

Build the characteristic polynomial

Subtract λ from the diagonal entries, calculate the determinant, and keep repeated factors visible.

Solve each eigenspace

For every eigenvalue, row reduce A - λI and describe all nonzero vectors in its null space.

Check multiplicity and verification

Compare algebraic and geometric multiplicities, then verify representative vectors with Av = λv.

Worked example

Find the eigenvalues and eigenvectors of A

A = [4213], det (A - λ I) = (4 - λ) (3 - λ) - 2

Form the characteristic determinant.

λ^{2} - 7 λ + 10 = (λ - 5) (λ - 2)

Factor the characteristic polynomial.

λ = 5 : v_{1} = [11], λ = 2 : v_{2} = [1 - 2]

Solve each null space for a representative eigenvector.

A v_{1} = 5 v_{1}, A v_{2} = 2 v_{2}

Verify both eigenpairs.

The eigenvalues are 5 and 2, with eigendirections spanned by [1, 1] and [1, -2].

Common mistakes to check

Stopping at the determinant

Eigenvalues are the roots of the characteristic equation, not the determinant expression itself.

Using the zero vector

The zero vector always satisfies Av = λv but is excluded from the definition of an eigenvector.

Ignoring repeated-root geometry

A repeated eigenvalue may have fewer independent eigenvectors than its algebraic multiplicity.

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Questions students ask

What is an eigenvalue?

It is the scalar by which a matrix scales a nonzero eigenvector without changing that vector's line of direction.

Are eigenvectors unique?

No. Any nonzero scalar multiple of an eigenvector lies in the same eigendirection.

Can a real matrix have complex eigenvalues?

Yes. A real matrix can have complex conjugate eigenvalue pairs.

Is every square matrix diagonalizable?

No. Diagonalization requires enough linearly independent eigenvectors to form a basis.