Eigenvalue & Eigenvector Calculator
Find the eigenvalues and a full eigenvector basis of a real 2 × 2 or 3 × 3 matrix.
Use integers, decimal dots or fractions such as -2/3. An empty cell is incomplete.
Paste a matrix
Matrix changed. Calculate to update the answer.
Diagonalizable over ℝ and ℂ · 2 independent eigenvectors.
λ₁ = 1
Eigenvector basis: (−1, 1)
Algebraic multiplicity 1 · eigenspace dimension 1.
λ₂ = 3
Eigenvector basis: (1, 1)
Algebraic multiplicity 1 · eigenspace dimension 1.
See A transform a vector
Drag the coral vector tip, use the arrow keys on it, or edit its coordinates. Try an eigenvector to see Av stay on its line.
Eigenvalues in the complex plane
A point represents an eigenvalue, not a real-space direction. Complex vectors are listed in their eigenvector bases above.
Characteristic equation and working
det(λI − A) = λ² − 4λ + 3 = 0
Display and numerical checks
Exact eigenvector identities are verified before display.
Decimal eigenvectors are approximate. The residual checks the full-precision unit vector, so rounded screen values may give a larger residual. A small residual alone is not proof of forward accuracy; close roots also undergo a separation check.
Worked examples
What the answer means
An eigenvector v is nonzero and satisfies Av = λv. Multiplying v by any nonzero scalar gives another eigenvector for the same λ. An eigenvalue can be zero: its eigenvectors collapse to the origin.
Algebraic multiplicity counts a root of the characteristic polynomial. Geometric multiplicity is the dimension of its eigenspace. A repeated eigenvalue can have several independent vectors, or fewer than its algebraic multiplicity. A matrix is diagonalizable over ℂ when the complete eigenbasis has as many independent vectors as the matrix size. Over ℝ it additionally needs all eigenvalues real.
With complex eigenvalues there may be no real invariant line, as in a quarter-turn rotation. Complex eigenvectors still exist; a plot of their real parts alone would not represent the eigenvectors.
Use the matrix calculator for determinants, inverses, RREF and general linear systems.
Input, precision and limits
Use a real square 2 × 2 or 3 × 3 matrix. Rows can be separated by newlines or semicolons; entries by spaces, tabs or commas. Decimal commas and expressions such as sqrt(2) are not accepted. Each coefficient has at most 24 digits and an exponent from −12 to 12; fractions combine two such numbers. Input is limited to 2,500 characters.
The characteristic polynomial, repeated roots and rational/complex-rational eigenspaces use exact integer fractions. General cubic and irrational roots use local high-precision arithmetic, with exact discriminant decisions, eigenvector residuals, trace/determinant and root-separation checks. This is a small-matrix learning tool, not a certified numerical linear algebra package. Extremely large intermediate fractions or unresolved conditioning produce an explicit error rather than a guessed answer. The 2D/3D drawing uses rounded coordinates and illustrates the calculation; the eigenbasis is listed numerically above.
Your matrix is calculated in this browser. It is not placed in a shared URL or sent in calculator analytics.
Find eigenvalues, full real or complex eigenvector bases and multiplicities. Explore the transformation and check diagonalization over real and complex numbers.
