Eigenvalue & Eigenvector Calculator
Enter a 2×2 or 3×3 matrix to get its eigenvalues and eigenvectors, with the characteristic polynomial posed, the exact solving method used (quadratic formula for 2×2; rational root search then a quadratic, or the trigonometric/Cardano method for the cubic, for 3×3), and a trace/determinant verification check.
3, 1
Eigenvalues of A
Characteristic polynomial — show the work
Eigenvectors (scaled so the largest component equals 1):
λ1 = 3 → v1 = [1, 1]
Show elimination steps
λ2 = 1 → v2 = [-1, 1]
Show elimination steps
Verification: trace & determinant checks
3×3 matrices are solved by a rational-root search first; if no rational root exists, by the trigonometric (real-root) or classic Cardano method for the cubic. How we calculate →
The characteristic polynomial: where eigenvalues come from
Every eigenvalue λ of a matrix A satisfies det(A − λI) = 0 — subtract λ from each diagonal entry, then find the values of λ that make the determinant zero. For a 2×2 matrix this collapses to a simple quadratic: λ² − trace(A)λ + det(A) = 0. For [[4, 1], [2, 3]] (trace = 7, det = 10): λ² − 7λ + 10 = 0, giving eigenvalues 5 and 2.
For a 3×3 matrix, the same determinant expands into a cubic: λ³ − trace(A)λ² + Mλ − det(A) = 0, where M is the sum of the three 2×2 "principal minors" left after deleting one row and the matching column. For [[2,0,0],[0,3,4],[0,4,9]] (trace = 14, M = 35, det = 22): λ³ − 14λ² + 35λ − 22 = 0, with eigenvalues 1, 11, 2.
Solving the cubic for 3×3 matrices: rational roots, then Cardano
Unlike the 2×2 quadratic, a cubic doesn't have a one-line formula that's practical to show step by step. The calculator above tries the most common real-world case first: a rational root search — testing simple fractions to see if any make the polynomial zero. If one is found, the cubic factors into (λ − root) × (a quadratic), and that remaining quadratic is solved the normal way.
If no rational root exists, the cubic is converted to a "depressed" form and solved either by the trigonometric method (when all three roots are real — this is the "casus irreducibilis" case, where the roots are real but genuinely irrational, like nested square roots that don't simplify) or by Cardano's classical formula (when there's exactly one real root and a complex-conjugate pair). Both methods are exact, not approximations — only the final decimal display is rounded.
Finding eigenvectors: elimination on (A − λI)
Once an eigenvalue λ is known, its eigenvector is any nonzero vector v solving (A − λI)v = 0. Because λ was chosen specifically to make (A − λI) singular, this system always has infinitely many solutions along a single direction — the calculator row-reduces (A − λI), picks the free variable, and back-substitutes to get one representative vector (scaled so its largest entry is 1).
For [[4, 1], [2, 3]] with λ = 5, (A − λI) = [[−1, 1], [2, −2]] — both rows say the same thing (x = y), so the eigenvector is any multiple of (1, 1). Multiplying: A × (1,1) = (5, 5) = 5 × (1,1) — confirming the vector really does just get scaled by λ, not rotated, which is the entire definition of an eigenvector.
What eigenvalues and eigenvectors actually mean
Geometrically, most vectors change both length AND direction when multiplied by a matrix. Eigenvectors are the special directions that only get stretched or shrunk — never rotated — and the eigenvalue is exactly the stretch factor. A negative eigenvalue flips the vector's direction (still along the same line); a complex eigenvalue means the transformation includes a genuine rotation with no real axis that stays fixed.
This is why eigenvalues show up everywhere from vibration analysis (natural frequencies) to Google's original PageRank algorithm (the dominant eigenvector of the link matrix) to statistics (principal component analysis finds the eigenvectors of a covariance matrix).
Frequently asked questions
How do you find the eigenvalues of a 2×2 matrix?
Solve λ² − trace(A)λ + det(A) = 0 with the quadratic formula, where trace(A) is the sum of the diagonal entries and det(A) is the determinant. For [[4, 1], [2, 3]]: trace = 7, det = 10, giving eigenvalues 5 and 2.
How do you find the eigenvalues of a 3×3 matrix?
Expand det(A − λI) into the cubic λ³ − trace(A)λ² + Mλ − det(A) = 0 (M = sum of the principal 2×2 minors), then solve it — first by testing for a rational root, and if none exists, by the trigonometric or Cardano method for the cubic.
How do you find an eigenvector once you have an eigenvalue?
Solve (A − λI)v = 0 by row reduction. Because λ makes (A − λI) singular by construction, you'll always find at least one free variable — set it to 1, back-substitute through the reduced rows, and the result is a valid eigenvector (any nonzero multiple of it is also valid).
What does it mean for eigenvalues to be complex?
Complex eigenvalues always come in conjugate pairs (a + bi and a − bi) for a real matrix, and mean the linear transformation includes a genuine rotation with no real direction left unchanged — a rotation matrix like [[0, −1], [1, 0]] has eigenvalues ±i and no real eigenvectors at all.
How can I check my eigenvalues are correct?
Two quick checks: the sum of all eigenvalues must equal the matrix's trace, and the product of all eigenvalues must equal its determinant. Both checks are shown automatically for every result above.
What's the difference between an eigenvalue and an eigenvector?
The eigenvector is a direction (a vector) that a matrix transformation leaves unrotated; the eigenvalue is the number that direction gets scaled by. They always come as a pair — an eigenvalue on its own doesn't tell you the direction, and an eigenvector without its eigenvalue doesn't tell you the scale factor.
Why doesn't every 3×3 matrix have three real eigenvalues?
Because the characteristic polynomial is a cubic, and a cubic with real coefficients always has at least one real root but can have either three real roots or one real root plus a complex-conjugate pair — it depends entirely on the specific numbers in the matrix.
Researched & verified by the Calcuris Data & Research Team. How we build and check our tools →