Enter a 3×3 Matrix
Use the matrix grid below, choose precision and tolerance, then compute the characteristic polynomial, eigenvalues, eigenspaces, and diagonalization test.
Example Data Table
| Example | Matrix | Expected Eigenvalues | Diagonalizable? | Reason |
|---|---|---|---|---|
| Symmetric repeated-root case | [4 1 1] [1 4 1] [1 1 4] |
3, 3, 6 | Yes | λ = 3 has a two-dimensional eigenspace, so three independent eigenvectors exist. |
| Distinct-root case | [5 2 0] [0 3 1] [0 0 1] |
1, 3, 5 | Yes | Three distinct real eigenvalues guarantee three independent eigenvectors. |
Formula Used
Characteristic equation: det(A − λI) = 0
Expanded cubic: λ³ − tr(A)λ² + σ₂λ − det(A) = 0
Second invariant: σ₂ = a₁₁a₂₂ + a₁₁a₃₃ + a₂₂a₃₃ − a₁₂a₂₁ − a₁₃a₃₁ − a₂₃a₃₂
Diagonalization test: If three independent eigenvectors exist, form P = [v₁ v₂ v₃] and D = diag(λ₁, λ₂, λ₃), then A = P D P⁻¹.
How to Use This Calculator
- Enter the nine values of your 3×3 matrix in the input grid.
- Choose the number of decimal places used in the displayed output.
- Set a numerical tolerance for repeated roots and near-singular eigenspaces.
- Select whether eigenvalues should appear in ascending or descending order.
- Press Diagonalize Matrix to generate the characteristic polynomial and eigenspace analysis.
- Read the status badge to confirm whether the matrix is diagonalizable over the real numbers.
- Use the CSV and PDF buttons to export the generated result block.
Spectrum and diagonalizability screening
For a 3×3 matrix, the first checkpoint is the eigenvalue spectrum. Distinct real eigenvalues support diagonalization because they produce independent eigenvectors. Repeated eigenvalues require a stricter test: the geometric multiplicity must equal the algebraic multiplicity. This calculator summarizes that decision in one result badge and then lists the eigenspace basis vectors that justify the conclusion numerically.
Characteristic polynomial and invariants
The characteristic polynomial condenses the matrix into a cubic equation, λ³ − tr(A)λ² + σ₂λ − det(A) = 0. Its coefficients come directly from the trace, determinant, and second invariant. Those values provide fast reasonableness checks. If the sum of the reported eigenvalues does not match the trace, or their product does not match the determinant, either the inputs or the interpretation need review.
Multiplicity and eigenspace quality
Repeated roots often create the difference between a clean diagonal form and a defective matrix. A 3×3 matrix with eigenvalues 6, 3, and 3 can still be diagonalizable if the λ = 3 eigenspace has dimension two. This page reports both multiplicities side by side so users can see whether the eigenspace is rich enough to support three independent directions.
Numerical tolerance and stability
Real computations are sensitive to rounding, especially when eigenvalues are close together or when matrix entries vary widely in scale. The tolerance setting helps classify nearly repeated roots and weak pivots during row reduction. Smaller tolerances are stricter but may exaggerate floating point noise. Larger tolerances are more forgiving but can merge values that should remain distinct in exact treatment.
Reconstruction audit and interpretation
A strong verification step is reconstruction: build P from eigenvectors, D from eigenvalues, compute P⁻¹, and compare A with PDP⁻¹. If the maximum entrywise difference stays small, the diagonalization is trustworthy. This audit is valuable in teaching, models, and data workflows because it shows not only that a factorization exists, but also that it behaves well in finite precision.
Practical use in coursework and modeling
This calculator supports classroom exercises, matrix method reviews, stability studies, and validation before symbolic work. Users can test examples, export reports, and compare matrices with distinct or repeated spectra. In practice, diagonalization simplifies powers of A, repeated transformations, linear differential systems, and proofs where eigenbasis structure matters.
FAQs
1. When is a 3×3 matrix diagonalizable?
A 3×3 matrix is diagonalizable when it has three linearly independent eigenvectors. Distinct real eigenvalues guarantee this, while repeated eigenvalues require enough independent vectors inside the repeated eigenspace.
2. What does the characteristic polynomial show?
It packages the matrix into a cubic equation whose roots are the eigenvalues. Its coefficients also reflect the trace, second invariant, and determinant, which help verify the calculation.
3. Why can repeated eigenvalues still be diagonalizable?
A repeated eigenvalue does not automatically block diagonalization. The key requirement is that its eigenspace dimension must match its algebraic multiplicity, giving enough independent eigenvectors overall.
4. What is the role of numerical tolerance?
Tolerance controls how the solver treats nearly repeated roots and very small pivot values. It helps stabilize row reduction and grouping decisions when decimal rounding affects the matrix.
5. Why is reconstruction useful?
Reconstruction checks whether P D P⁻¹ reproduces the original matrix closely. A small error confirms that the reported eigenvectors, inverse, and diagonal matrix are numerically consistent.
6. Can this calculator handle complex diagonalization?
This version focuses on real diagonalization. If the cubic produces only one real eigenvalue and a complex conjugate pair, the tool reports that real diagonalization is unavailable.