3x3 Eigenvalues Calculator

Analyze any 3x3 matrix using clear matrix invariants. View real and complex roots with clarity. Export summaries, compare cases, and verify learning with examples.

Enter Your 3×3 Matrix

Use the responsive input grid below.

• Nine entries are required.
• Complex eigenvalues appear automatically.
• Exports become available after calculation.

Example Data Table

This sample matrix has three real eigenvalues.

Example Matrix Trace Determinant Expected Eigenvalues
Worked Example [4, 1, 0]
[1, 4, 0]
[0, 0, 2] 		10 30 5, 3, 2

Use the Load Example button to insert this matrix instantly.

Formula Used

1) Characteristic Polynomial

For a 3×3 matrix A, eigenvalues satisfy det(λI - A) = 0.

2) Expanded Cubic Form

λ³ - tr(A)λ² + I₂λ - det(A) = 0

3) Trace

tr(A) = a₁₁ + a₂₂ + a₃₃

4) Second Invariant

I₂ = a₁₁a₂₂ + a₁₁a₃₃ + a₂₂a₃₃ - a₁₂a₂₁ - a₁₃a₃₁ - a₂₃a₃₂

5) Determinant Relation

The product of all eigenvalues equals det(A).

6) Verification Identities

The sum of eigenvalues equals the trace. The sum of pairwise products equals I₂.

How to Use This Calculator

Step 1

Enter all nine numbers from your 3×3 matrix.

Step 2

Choose the decimal precision and output order.

Step 3

Press Calculate Eigenvalues to solve the cubic equation.

Step 4

Review the table, notes, graph, and export options.

Tip: Symmetric matrices usually produce real eigenvalues. Non-symmetric matrices may produce complex-conjugate pairs.

FAQs

1) What does this calculator compute?

It computes the eigenvalues of any 3×3 matrix, along with the trace, determinant, second invariant, characteristic polynomial, and a visual root plot.

2) Can it show complex eigenvalues?

Yes. If the matrix produces non-real roots, the calculator displays their real and imaginary parts and plots them on the complex plane.

3) Why are trace and determinant included?

They help verify the answer. The sum of eigenvalues equals the trace, and the product of eigenvalues equals the determinant.

4) What does the second invariant mean?

It is the coefficient of the linear term in the characteristic polynomial and equals the sum of pairwise eigenvalue products.

5) Will symmetric matrices always give real eigenvalues?

Yes, in exact mathematics. Very tiny imaginary parts can appear only because of floating-point rounding during numerical computation.

6) What happens when the determinant is zero?

At least one eigenvalue becomes zero. That means the matrix is singular and cannot be inverted.

7) Can I export the results?

Yes. After calculation, you can download a CSV summary or generate a PDF report with the matrix and eigenvalue table.

8) Is this suitable for study and verification?

Yes. The calculator includes formulas, a worked example, interpretation notes, and exportable results for homework checks or teaching demonstrations.

Related Calculators

3x3 eigenvectorsmatrix eigenvectorsleft eigenvectorscovariance eigenvalueseigen decomposition stepseigenvalue condition numberunitary eigenvalueseigenvalues findereigenvector solversymmetric eigenvalues

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.