Calculator · Engineering Mathematics
Eigenvalues & Eigenvectors Calculator
Solve A v = λ v for a real square matrix. Complex conjugate pairs keep their imaginary parts, repeated eigenvalues are reported with their geometric multiplicity, and every eigenpair is checked against the original matrix.
Matrix A
One row per line; entries separated by spaces or commas. Scientific notation such as 1.5e-3 is accepted. At most 10 × 10; ragged rows and blank entries are refused, never repaired.
What Anemodia read
| 0 | -1 |
| 1 | 0 |
Results
| Size | 2 × 2 |
|---|---|
| Symmetric | No |
| Distinct eigenvalue positions | 2 |
| Spectral radius max|λ| | 1 |
| ‖A‖₂ | 1 |
| Diagonalisable | Yes — every eigenvalue has a complete eigenspace |
| λ | |λ| | Type | Eigenvector v (‖v‖₂ = 1) | ‖Av − λv‖₂ | Relative residual | Status | ||
|---|---|---|---|---|---|---|---|---|
| 0 + 1i | 1 | complex | Eigenvector for λ = 0 + 1i
| 0 | 0 | verified | ||
| 0 − 1i | 1 | complex | Eigenvector for λ = 0 − 1i
| 0 | 0 | verified |
Eigenvalue plane
Plotted eigenvalues
- λ = 0 + 1i0 + 1i
- λ = 0 − 1i0 − 1i
Multiplicity
- λ = 0 + 1i — algebraic multiplicity 1, geometric multiplicity 1. Simple eigenvalue: one independent direction.
- λ = 0 − 1i — algebraic multiplicity 1, geometric multiplicity 1. Simple eigenvalue: one independent direction.
Continue
The matrix continues into the Lab's Eigen workspace, alongside its rank, conditioning and factorisation. No right-hand side is carried over, because this calculator does not have one.
Assumptions & validity
- · A must be real, square and at most 10 × 10.
- · Eigenvalues are computed by the QR algorithm, not by finding roots of the characteristic polynomial — that route is numerically unreliable beyond very small matrices.
- · Complex conjugate pairs keep their imaginary parts. The eigenvector is reconstructed as v = v_r ± i v_i and reported as a complex vector, never as two unexplained real columns.
- · Every eigenvector is normalised to ‖v‖₂ = 1 in the complex Euclidean norm. Sign and phase remain arbitrary: v and −v, or e^{iθ}v, are the same eigenvector.
- · Every eigenpair is validated against the ORIGINAL matrix through ‖Av − λv‖₂; a pair that fails is marked unverified rather than presented as an eigenvector.
- · For a repeated real eigenvalue the geometric multiplicity is nullity(A − λI). For a repeated COMPLEX eigenvalue V1 does not decide it, and says so instead of claiming a complete basis.
- · Eigenvalues closer than a scale-aware numerical clustering tolerance are treated as numerically repeated for multiplicity analysis. This is a heuristic, not an exact multiplicity test: the solver returns floating-point approximations, and eigenvalues that close are indistinguishable at working precision. The tolerance scales with A, so replacing A by cA does not change the multiplicity interpretation.
Technical reference
A v = λ v, v ≠ 0
det(A − λ I) = 0 (the defining relation, not the method used here)
‖A v − λ v‖₂ and ‖A v − λ v‖₂ / ((‖A‖₂ + |λ|)‖v‖₂)
geometric multiplicity = nullity(A − λ I) ≤ algebraic multiplicity
- λ
- eigenvalue, possibly complex even for a real matrix
- v
- eigenvector, defined only up to a non-zero scale or phase
- ‖A‖₂
- induced 2-norm σmax, used to make the residual relative
The characteristic equation det(A − λI) = 0 defines the eigenvalues, but it is not how they are computed: the coefficients of that polynomial are extremely sensitive, so its roots are a poor way to find a spectrum. The QR algorithm works on the matrix directly.
A real matrix can have complex eigenvalues, and they always come in conjugate pairs λ = a ± ib with conjugate eigenvectors. Discarding the imaginary part would not simplify the answer; it would replace it with a wrong one.
A repeated eigenvalue does NOT guarantee enough independent eigenvectors. When the geometric multiplicity is smaller than the algebraic one the matrix is defective and cannot be diagonalised — the classic example is [[1,1],[0,1]], which has λ = 1 twice but only one eigenvector direction.
Eigenvectors are never compared here by raw component equality: two correct answers can differ by a sign or a complex phase. The residual and the direction are what matter.