Enter Matrix Values
Choose a 2×2 or 3×3 square matrix. Then compute eigenvalue multiplicities, eigenspace dimensions, and diagonalizability checks.
Formula Used
For a square matrix A, eigenvalues come from:
p(λ) = det(A - λI)
The algebraic multiplicity of an eigenvalue is the number of times that eigenvalue appears as a root of the characteristic polynomial.
The geometric multiplicity equals the dimension of the eigenspace:
GM(λ) = dim Ker(A - λI) = n - rank(A - λI)
A matrix is diagonalizable when the total number of linearly independent eigenvectors equals the matrix size. In practice, this means the sum of geometric multiplicities must equal n.
How to Use This Calculator
- Select whether your matrix is 2×2 or 3×3.
- Enter every matrix entry in the responsive input grid.
- Adjust precision and tolerance if you want tighter grouping.
- Press Calculate Multiplicities.
- Read the characteristic polynomial, eigenvalues, algebraic multiplicity, geometric multiplicity, and diagonalizability result above the form.
- Use the CSV or PDF buttons to export the computed result.
- Check the Plotly chart to compare multiplicity values visually.
Example Data Table
| Example Matrix | Eigenvalues | Algebraic Multiplicity | Geometric Multiplicity | Diagonalizable? |
|---|---|---|---|---|
| [[2, 0], [0, 3]] | 2, 3 | 1, 1 | 1, 1 | Yes |
| [[2, 1], [0, 2]] | 2 | 2 | 1 | No |
| [[3, 0, 0], [0, 3, 0], [0, 0, 1]] | 3, 1 | 2, 1 | 2, 1 | Yes |
| [[4, 1, 0], [0, 4, 1], [0, 0, 4]] | 4 | 3 | 1 | No |
Frequently Asked Questions
1) What is algebraic multiplicity?
It is the number of times an eigenvalue repeats as a root of the characteristic polynomial. A repeated root has algebraic multiplicity greater than one.
2) What is geometric multiplicity?
It is the number of linearly independent eigenvectors associated with one eigenvalue. Equivalently, it is the dimension of the eigenspace for that eigenvalue.
3) Can geometric multiplicity exceed algebraic multiplicity?
No. Geometric multiplicity is always at least one for an eigenvalue, but it can never be larger than the algebraic multiplicity.
4) When is a matrix diagonalizable?
A matrix is diagonalizable when it has enough independent eigenvectors to form a basis. The sum of geometric multiplicities must equal the matrix dimension.
5) Why does a repeated eigenvalue sometimes fail diagonalization?
A repeated eigenvalue may have too few independent eigenvectors. Then geometric multiplicity is smaller than algebraic multiplicity, making the matrix defective and not diagonalizable.
6) What does the tolerance field do?
Tolerance controls how closely two computed roots must match before the calculator groups them as the same eigenvalue. Smaller tolerance gives stricter grouping.
7) Does this tool support decimal entries?
Yes. You can enter integers or decimals. The calculator then computes the characteristic polynomial numerically and reports multiplicities using the chosen precision.
8) What if my matrix has complex eigenvalues?
The tool lists complex roots. For simple complex roots, geometric multiplicity is shown as one over complex scalars. Repeated complex roots are flagged with a note.