Study defective eigenvalues from matrices and repeated roots. Check eigenspace dimensions, nullity, and diagonalization limits. Review steps, download reports, and inspect plotted eigenvalue structure.
A defective eigenvalue appears when a repeated eigenvalue does not produce enough independent eigenvectors. This matters because diagonalization depends on matching algebraic multiplicity with geometric multiplicity. When the geometric multiplicity is smaller, the matrix needs at least one Jordan block larger than one.
This calculator accepts real 2 x 2 and 3 x 3 matrices. It builds the characteristic polynomial, estimates the eigenvalues, groups repeated roots, and tests the eigenspace dimension for repeated real roots. The result section then reports whether the matrix is defective and whether diagonalization over C is likely.
The page is useful for linear algebra homework, control systems checks, repeated-root studies, and quick verification before doing full symbolic Jordan form work by hand. You can also export the result summary, inspect the plotted eigenvalue locations, and compare your matrix with the included example cases.
For a square matrix A, eigenvalues satisfy det(A - λI) = 0. The algebraic multiplicity of λ is the number of times it appears as a root. The geometric multiplicity of λ is dim Null(A - λI).
A repeated eigenvalue is defective when:
geometric multiplicity < algebraic multiplicity
For 2 x 2 matrices, the characteristic polynomial is:
λ² - tr(A)λ + det(A) = 0
For 3 x 3 matrices, the characteristic polynomial is:
λ³ - tr(A)λ² + Sλ - det(A) = 0
where S = 1/2[(tr(A))² - tr(A²)].
| Case | Matrix | Repeated Eigenvalue | Expected Outcome |
|---|---|---|---|
| Simple defective 2 x 2 | [[5, 1], [0, 5]] | 5 | Defective, one eigenvector only |
| Repeated but not defective 2 x 2 | [[3, 0], [0, 3]] | 3 | Not defective, two independent eigenvectors |
| Jordan style 3 x 3 | [[2, 1, 0], [0, 2, 1], [0, 0, 2]] | 2 | Defective, one eigenvector |
| Diagonal 3 x 3 | [[4, 0, 0], [0, 4, 0], [0, 0, 1]] | 4 | Not defective, full eigenspace for 4 |
Numerical computations can blur very close roots. If your matrix is near defective, try a smaller tolerance and compare the result. For symbolic classroom work, always confirm borderline cases by hand using the nullspace of A - λI.
An eigenvalue is defective when its algebraic multiplicity exceeds its geometric multiplicity. In practice, the repeated root produces too few independent eigenvectors, so the matrix cannot be fully diagonalized around that eigenvalue.
No. A repeated eigenvalue can still be nondefective if the eigenspace has enough independent eigenvectors to match the algebraic multiplicity. The calculator checks both multiplicities before labeling the matrix defective.
Numerical roots and row reduction can produce tiny rounding noise. Tolerance tells the calculator when values are effectively zero, and when nearby roots should be treated as the same repeated eigenvalue.
Yes. This page analyzes real 2 x 2 and 3 x 3 matrices. It computes the characteristic polynomial, estimates eigenvalues, checks eigenspace dimension for repeated real roots, and summarizes whether defect behavior appears.
It is the dimension of the nullspace of A - λI. That dimension equals the number of independent eigenvectors associated with the eigenvalue and determines whether diagonalization remains possible.
A matrix fails to diagonalize when at least one repeated eigenvalue has too few independent eigenvectors. That shortfall creates a Jordan block larger than one, which this calculator flags as defective.
The Plotly graph places computed eigenvalues on the complex plane using real and imaginary parts. Repeated roots appear at the same location, making clustering and spectrum structure easier to inspect.
For 2 x 2 matrices, roots come from the quadratic formula. For 3 x 3 matrices, roots are computed numerically from the cubic formula, so very close cases may depend slightly on tolerance and rounding.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.