Math
Eigenvalues and Eigenvectors Calculator
Enter a square matrix to get its characteristic polynomial, every eigenvalue, and a basis of eigenvectors for each one.
Rational eigenvalues are exact; 2 x 2 and quadratic leftovers give exact square-root or complex forms; anything else is solved numerically and labelled approximate. The panel also lists algebraic and geometric multiplicities, checks trace and determinant against the eigenvalues, and says whether A can be diagonalized.
One row per line. Must be square. Try a symmetric matrix for real eigenvalues, or a rotation like [0 -1; 1 0] for complex ones.
Try an example
Result
Eigenvalues
—
- Characteristic polynomial
- —
- Trace
- —
- Determinant
- —
- Diagonalizable?
- —
Student quick launch
Grade planning, algebra checks, and formulas students reach for most.
Study path
Use this calculator with
Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
What eigenvalues and eigenvectors are
An eigenvector of a square A rectangular grid of numbers arranged in rows and columns. A is a nonzero A quantity with both a size and a direction, often drawn as an arrow. v that A only stretches: Av = λv. The stretch factor λ is the A number saying how much a matrix stretches vectors along one special direction.. Eigenvalues are the roots of the characteristic polynomial det(λI - A); eigenvectors for a given λ are the nonzero solutions of (A - λI)v = 0, i.e. the null space of A - λI.
How the calculator finds them
- Builds the characteristic polynomial exactly with the Faddeev-LeVerrier recurrence, so its The number multiplying a variable, like the 3 in 3x. are exact fractions.
- Finds every rational root exactly (rational root test) and divides it out, repeating for repeated roots.
- Solves any leftover quadratic exactly, giving square-root forms such as 1/2 + (1/2)√5 or complex pairs such as 1 ± 2i.
- Solves a leftover cubic or higher numerically and marks those eigenvalues approximate.
- Computes eigenvectors as a basis of the null space of A - λI: exactly for rational λ, numerically otherwise.
How to read the answer
| Panel item | Meaning |
|---|---|
| Algebraic multiplicity | How many times λ is a root of the characteristic polynomial |
| Geometric multiplicity | How many independent eigenvectors λ has (dimension of its eigenspace) |
| Diagonalizable | Yes when geometric = algebraic for every λ, so A = PDP⁻¹ with D diagonal |
| Trace = sum of λ, det = product of λ | Quick checks that the eigenvalues are right |
| Complex pair a ± bi | A rotates as well as scales; no real eigenvector for those λ |
Common mistakes
- Forgetting that eigenvectors are only defined up to scale: (1, 2) and (2, 4) are the same eigenvector direction. The calculator scales exact eigenvectors to small integers.
- Using the zero vector as an eigenvector. It always satisfies Av = λv, so it is excluded by definition.
- Assuming every matrix has n independent eigenvectors. A repeated eigenvalue can have fewer (a defective matrix).
- Expecting real eigenvalues from a non-symmetric matrix. Rotation-like matrices have complex eigenvalues.
How to type a matrix into this calculator
- Put each row on its own line. Separate the entries in a row with spaces or commas: 2 1 1 or 2, 1, 1.
- Fractions such as 1/3 and decimals such as 0.25 are accepted and kept exact, so 1/3 stays 1/3 instead of becoming 0.3333.
- Every row must have the same number of entries. A row with a missing entry produces a clear error instead of a wrong answer.
- The largest size supported is 8 x 8. Brackets and semicolons are ignored, so [1 2; 3 4] works too.
Worked examples
Symmetric 2 x 2
A = [2 1; 1 2] → λ = 1, 3 with eigenvectors (-1, 1) and (1, 1)
Eigenvalues
1; 3
Irrational eigenvalues (Fibonacci matrix)
A = [1 1; 1 0] → λ = (1 ± √5)/2
Eigenvalues
1/2 - (1/2)√5 ≈ -0.618; 1/2 + (1/2)√5 ≈ 1.618
Complex eigenvalues (rotation by 90°)
A = [0 -1; 1 0] → λ = ±i
Eigenvalues
-i; i
3 x 3 with integer eigenvalues
A = [2 0 0; 0 3 4; 0 4 9] → λ = 1, 2, 11
Eigenvalues
1; 2; 11
Defective matrix
A = [1 1; 0 1] has λ = 1 twice but only one eigenvector
Eigenvalues
1
Singular matrix has eigenvalue 0
A = [1 2; 2 4] → λ = 0, 5
Eigenvalues
0; 5
Not square
A 2 x 3 matrix has no eigenvalues
Eigenvalues
Error
Frequently asked questions
What does an eigenvalue tell you?+
It is the factor by which the matrix stretches its eigenvector. λ > 1 stretches, 0 < λ < 1 shrinks, λ < 0 flips, λ = 0 collapses that direction (and makes the matrix singular).
How do I find eigenvalues of a 2 x 2 matrix by hand?+
Solve λ² - (trace)λ + det = 0 with the quadratic formula. For [2 1; 1 2] that is λ² - 4λ + 3 = 0, giving λ = 1 and λ = 3.
Why did I get complex eigenvalues?+
The characteristic polynomial has a negative discriminant. Geometrically the matrix rotates vectors, so no real vector keeps its direction. Symmetric matrices never do this; their eigenvalues are always real.
What is the difference between algebraic and geometric multiplicity?+
Algebraic multiplicity counts how often λ appears as a root of the characteristic polynomial. Geometric multiplicity is the number of independent eigenvectors for λ. Geometric is never larger than algebraic; when it is smaller the matrix is defective.
Can eigenvectors be scaled?+
Yes. Any nonzero multiple of an eigenvector is also an eigenvector for the same λ. Different tools may show (1, 1) or (0.707, 0.707); both are correct.
How do eigenvalues relate to trace and determinant?+
The sum of the eigenvalues (with multiplicity) equals the trace and their product equals the determinant. The panel shows both as a check.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
Related calculators
Characteristic Polynomial Calculator
Find the characteristic polynomial det(λI - A) of any square matrix up to 8 x 8 with exact coefficients. Shows the Faddeev-LeVerrier steps, trace and determinant checks, the factored form, the roots (eigenvalues), and a Cayley-Hamilton verification.
Matrix Determinant Calculator
Find the determinant of any square matrix up to 8 x 8 with exact fractions. Shows cofactor expansion along any row or column for 3 x 3 (with a rule-of-Sarrus check), row-reduction steps for larger matrices, and tells you whether the matrix is invertible.
Null Space Calculator
Find a basis for the null space (kernel) of any matrix up to 8 x 8 with exact integer-scaled vectors, plus the column space and row space bases, rank, nullity, free variables, and every row operation.
Matrix Trace Calculator
Find the trace of a square matrix up to 8 x 8 (the sum of its diagonal entries) with exact fractions. Shows the running sum, the eigenvalue check trace = λ1 + … + λn, the determinant, tr(A²), and the trace properties.
Matrix Power Calculator
Raise a square matrix up to 8 x 8 to any integer power from -30 to 30 with exact fractions. Shows each intermediate power, handles A^0 = I and negative powers through the inverse, and checks det(A^k) = det(A)^k.
Quadratic Formula Calculator
Solve ax^2 + bx + c = 0 with exact simplified radicals, reduced fractions, or complex roots, plus discriminant, vertex, factored and vertex forms, a graph, and every step.
Last updated: September 4, 2026