3x3 Eigenvectors Calculator

Enter Matrix Values

Example Data Table

Example Matrix	Eigenvalues	Representative Eigenvectors	Notes
[[4, 1, 1], [1, 4, 1], [1, 1, 4]]	6, 3, 3	λ=6: [1,1,1], λ=3: [1,-1,0], [1,1,-2]	Repeated eigenvalue with a two-dimensional eigenspace. The matrix is diagonalizable.
[[2, 0, 0], [0, 3, 0], [0, 0, 5]]	5, 3, 2	[0,0,1], [0,1,0], [1,0,0]	Diagonal matrices have immediate eigenvectors aligned with coordinate axes.

Formula Used

For a 3×3 matrix A, the eigenvalues come from the characteristic equation:

det(λI - A) = λ³ - tr(A)λ² + s₂λ - det(A) = 0

Here tr(A) is the trace, det(A) is the determinant, and s₂ is the sum of principal 2×2 minors.

After finding each eigenvalue λ, the calculator solves:

(A - λI)v = 0

The nonzero solutions form the eigenspace. Basis vectors are normalized for cleaner reporting, and complex values are supported when the matrix produces non-real eigenvalues.

How to Use This Calculator

Enter the nine values of your 3×3 matrix.

Choose the decimal precision for displayed results.

Adjust tolerance when near-zero numerical noise needs control.

Click Calculate Eigenvectors to compute the full result.

Review the characteristic polynomial, root type, and diagonalizability.

Inspect each eigenvalue, multiplicity, and eigenspace basis vector.

Use the graph to see where eigenvalues lie in the complex plane.

Download CSV or PDF copies for reporting or study notes.

Frequently Asked Questions

1) What does this calculator find?

It computes eigenvalues, eigenspace basis vectors, the characteristic polynomial, trace, determinant, multiplicities, and diagonalizability checks for any numeric 3×3 matrix.

2) Can it handle complex eigenvalues?

Yes. If the characteristic polynomial has non-real roots, the calculator displays complex eigenvalues and corresponding complex eigenvectors in standard a + bi form.

3) Why are repeated eigenvalues shown once?

Repeated roots are grouped into one eigenspace entry. Their algebraic multiplicity shows how many times the root appears, while the geometric multiplicity shows the basis size.

4) What is algebraic multiplicity?

Algebraic multiplicity is the number of times an eigenvalue repeats as a root of the characteristic polynomial.

5) What is geometric multiplicity?

Geometric multiplicity is the number of linearly independent eigenvectors attached to one eigenvalue. It equals the dimension of that eigenspace.

6) Why can eigenvectors look different from textbook answers?

Any nonzero scalar multiple of an eigenvector is still valid. This calculator normalizes vectors, so your answer may differ in scale but represent the same direction.

7) When is a 3×3 matrix diagonalizable?

A matrix is diagonalizable when it has three linearly independent eigenvectors. This tool checks that through the total geometric multiplicity across all eigenvalues.

8) What does the graph show?

The plot marks eigenvalues on the complex plane. The horizontal axis is the real part, and the vertical axis is the imaginary part.