Calculator
Formula Used
The calculator first builds the characteristic polynomial det(λI − A).
Algebraic multiplicity is the number of times an eigenvalue appears as a root of that polynomial.
Geometric multiplicity is the nullity of A − λI. Nullity equals n − rank(A − λI).
For a 2 x 2 matrix, the polynomial is λ² − tr(A)λ + det(A).
For a 3 x 3 matrix, the polynomial is λ³ − tr(A)λ² + s₂λ − det(A), where s₂ = ((tr(A))² − tr(A²)) / 2.
How to Use This Calculator
- Select a 2 x 2 or 3 x 3 matrix size.
- Enter every visible matrix entry.
- Keep the tolerance small for stable clustering of repeated roots.
- Click Calculate Multiplicity.
- Read the characteristic polynomial, eigenvalues, algebraic multiplicities, and geometric multiplicities.
- Check the diagonalizability note to see whether the matrix has enough independent eigenvectors.
- Use the CSV or PDF button to save the result summary.
Example Data Table
| Example Matrix | Key Eigenvalues | Multiplicity Summary | Interpretation |
|---|---|---|---|
| [[4,1,0],[0,4,0],[0,0,2]] | 4, 2 | λ=4 → AM 2, GM 1; λ=2 → AM 1, GM 1 | Not diagonalizable. One repeated root lacks enough eigenvectors. |
| [[3,0,0],[0,3,0],[0,0,1]] | 3, 1 | λ=3 → AM 2, GM 2; λ=1 → AM 1, GM 1 | Diagonalizable. The repeated eigenvalue has a full eigenspace. |
| [[0,-1],[1,0]] | i, -i | Each eigenvalue has AM 1 and GM 1 | Diagonalizable over complex numbers. |
Algebraic and Geometric Multiplicity in Linear Algebra
Why these multiplicities matter
Algebraic and geometric multiplicity are core ideas in eigenvalue analysis. They describe how a matrix behaves near a repeated root. Algebraic multiplicity counts how often an eigenvalue appears in the characteristic polynomial. Geometric multiplicity counts the independent eigenvectors attached to that eigenvalue. These values shape diagonalization, Jordan structure, stability work, and matrix powers.
What the calculator evaluates
This calculator studies real square matrices of order two or three. It forms the characteristic polynomial. Then it solves for eigenvalues. After that, it measures the nullity of A minus λI. That nullity gives the geometric multiplicity. The comparison between both multiplicities shows whether a repeated eigenvalue is complete or defective.
How to read the output
If algebraic multiplicity equals geometric multiplicity, the eigenspace is large enough for that eigenvalue. If geometric multiplicity is smaller, the matrix is defective at that root. A defective repeated eigenvalue often means the matrix is not diagonalizable. That result matters in differential equations, control, numerical methods, and matrix exponential problems.
Common learning pattern
Students often find repeated eigenvalues confusing. A double root does not guarantee two eigenvectors. That is the key distinction. One matrix may have a repeated eigenvalue and still diagonalize cleanly. Another may fail because the eigenspace has lower dimension. This calculator makes that contrast visible with direct multiplicity counts and an interpretation note.
Practical value
Use this tool to verify homework, build intuition, and inspect matrix structure quickly. It is also helpful when checking Jordan form readiness, similarity transformations, and modal decomposition steps. Because the results include the characteristic polynomial, nullity logic, and diagonalizability status, the page supports both quick computation and concept review in one place.
FAQs
1. What is algebraic multiplicity?
Algebraic multiplicity is the number of times an eigenvalue appears as a root of the characteristic polynomial. A repeated root has algebraic multiplicity greater than one.
2. What is geometric multiplicity?
Geometric multiplicity is the dimension of the eigenspace for one eigenvalue. It equals the nullity of A minus λI.
3. Can geometric multiplicity exceed algebraic multiplicity?
No. Geometric multiplicity is always less than or equal to algebraic multiplicity. That rule is fundamental in linear algebra.
4. When is a matrix diagonalizable?
A matrix is diagonalizable over complex numbers when every eigenvalue has enough independent eigenvectors. In practice, that means geometric multiplicity matches algebraic multiplicity for each eigenvalue.
5. Why can a repeated eigenvalue fail diagonalization?
A repeated eigenvalue can still have too small an eigenspace. When that happens, the matrix is defective and cannot be diagonalized fully.
6. Does this calculator handle complex eigenvalues?
Yes. The solver can report complex eigenvalues and their multiplicities. The diagonalizability summary is interpreted over complex numbers.
7. Why is tolerance included?
Tolerance helps group nearly identical numerical roots. That is useful when a repeated eigenvalue is found through floating point computation.
8. What size matrices can I test here?
This version supports 2 x 2 and 3 x 3 matrices. That keeps the workflow simple while still covering many teaching and practice cases.