Eigenvalue & Eigenvector Calculator
Characteristic polynomial, eigenvalues and unit eigenvectors for 2×2 and 3×3 matrices — repeated roots reported honestly, complex pairs included, and every result checked against the trace and determinant identities.
The default 2×2 is the textbook case: λ = 5 with v = (1, 1) and λ = 2 with v = (1, −2).
How we got it — the example, step by step
- det(A − λI) = (4 − λ)(3 − λ) − 2 = λ² − 7λ + 10 = (λ − 5)(λ − 2).
- λ = 5: (A − 5I)v = 0 gives −x + y = 0, so v = (1, 1). λ = 2: 2x + y = 0, so v = (1, −2). The page quotes the unit-length versions.
Check it backwards
Questions people also ask
Why is my eigenvector's sign or scale different from my textbook?
Eigenvectors are defined only up to a nonzero multiple — (1, 1), (2, 2) and (−1, −1) are the same direction. This page quotes unit-length vectors so answers can be compared; a multiple of the book's answer is the same answer.
What if the eigenvalues are complex?
Then A rotates as it scales and no real direction survives — the rotation matrix [[0, −1], [1, 0]] gives λ = ±i. For 2×2 you get the complex eigenvectors too; for 3×3 the complex eigenvalues are reported and the page says plainly that their vectors are not shown.
Are 4×4 matrices supported?
No — 2×2 and 3×3 only. A quartic characteristic polynomial has no tidy classroom route, and a numerical 4×4 eigensolver deserves its own page rather than a silent approximation here.