What is an eigenvalue really?

The λ satisfying Av = λv. The matrix acting on that eigenvector only stretches it by λ without turning it; a negative λ flips the direction and |λ| is the stretch factor.

Why is an eigenvector not unique?

If v is an eigenvector then so is any non-zero multiple kv, because A(kv) = λ(kv). An eigenvector really denotes a direction, so the tool normalises it to have largest component 1 for easy comparison.

Can eigenvalues be complex?

Yes. The rotation matrix [[0,−1],[1,0]] has eigenvalues ±i because no direction in the plane is merely stretched by a rotation. Complex eigenvalues are listed as a ± bi.

Which matrices are diagonalizable?

Those with n linearly independent eigenvectors. Distinct eigenvalues always suffice; with a repeated eigenvalue it depends on the dimension of its eigenspace. A symmetric matrix is always diagonalizable and its eigenvectors are orthogonal.

How do the trace and determinant relate to the eigenvalues?

The eigenvalues sum to the trace and multiply to the determinant. Both make quick sanity checks on a computed set of eigenvalues.

λEigenvalue Calculator - Eigenvalues and Eigenvectors of 2×2 and 3×3 Matrices

Type in a matrix and get the characteristic polynomial and all of its eigenvalues. Every real eigenvalue comes with its eigenvector and an Av = λv check, alongside the trace, the determinant and a verdict on diagonalizability. 2×2 uses the quadratic formula, 3×3 solves the cubic, and complex eigenvalues are reported as such.

Free · No sign-up · Runs in your browserLast updated: 2026-10-26

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How to use Eigenvalues

1Choose the size (2×2 or 3×3)
2Fill in the entries of the matrix
3Read the characteristic polynomial and the eigenvalues
4Check each eigenvector and its Av = λv verification

FAQ

What is an eigenvalue really?: The λ satisfying Av = λv. The matrix acting on that eigenvector only stretches it by λ without turning it; a negative λ flips the direction and |λ| is the stretch factor.
Why is an eigenvector not unique?: If v is an eigenvector then so is any non-zero multiple kv, because A(kv) = λ(kv). An eigenvector really denotes a direction, so the tool normalises it to have largest component 1 for easy comparison.
Can eigenvalues be complex?: Yes. The rotation matrix [[0,−1],[1,0]] has eigenvalues ±i because no direction in the plane is merely stretched by a rotation. Complex eigenvalues are listed as a ± bi.
Which matrices are diagonalizable?: Those with n linearly independent eigenvectors. Distinct eigenvalues always suffice; with a repeated eigenvalue it depends on the dimension of its eigenspace. A symmetric matrix is always diagonalizable and its eigenvectors are orthogonal.
How do the trace and determinant relate to the eigenvalues?: The eigenvalues sum to the trace and multiply to the determinant. Both make quick sanity checks on a computed set of eigenvalues.