Eigenvalue Calculator
Complex eigenvalues named, not refused.
Work out Eigenvalue. Complex eigenvalues named, not refused. States the assumption instead of hiding it.
Eigenvalues
4, 2, 1
All real
An eigenvalue is the factor by which the matrix stretches its own eigenvector — the directions the transformation leaves pointing the same way. The sum of the eigenvalues is always the trace and their product is always the determinant, which makes both a free check on the arithmetic. Complex eigenvalues are reported rather than refused. They mean the transformation rotates: a pure rotation matrix has eigenvalues ±i and no real eigenvector at all, which is correct rather than a failure. This stops at 3×3 deliberately. Beyond that the characteristic polynomial is a numerically terrible route — its roots are wildly sensitive to its coefficients — and the right answer is a QR iteration, which is a different piece of software from a calculator page.
How the Eigenvalue Calculator works
Eigenvalues of a 2×2 or 3×3 matrix from the characteristic polynomial, solved exactly. Complex eigenvalues are reported rather than dismissed — a rotation matrix has them, and that is correct rather than a failure.
Also known as: find eigenvalues of a matrix · characteristic equation solver · eigenvalues 2x2 calculator · what are complex eigenvalues
Frequently asked questions
What is an eigenvalue?
The factor by which a matrix stretches its own eigenvector — the special directions the transformation leaves pointing the same way. Everything else gets rotated as well as scaled.
How do I check the answer?
The eigenvalues must sum to the trace and multiply to the determinant. Both are shown here, and both are free checks on the arithmetic that hold for any square matrix.
What do complex eigenvalues mean?
The transformation rotates. A pure rotation matrix has eigenvalues ±i and no real eigenvector at all, because no real direction survives a rotation unchanged. They are meaningful, not an error.
Why does this stop at 3×3?
Because beyond that the characteristic polynomial is a numerically terrible route — its roots are extraordinarily sensitive to its coefficients. Larger matrices need a QR iteration, which is a different piece of software from a calculator page.
What are eigenvalues used for?
Principal component analysis, stability analysis of differential equations, vibration modes in engineering, and Google's original PageRank. Wherever a system has natural modes, eigenvalues describe them.
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