Eigenvalue and Eigenvector Calculator

Compute eigenvalues and eigenvectors for 2x2 and 3x3 matrices. Includes step-by-step formulas, examples, and notes on numerical stability.

Eigenvalue & Eigenvector

Results

Eigenvalues

[3.000000, 1.000000]

Eigenvectors (normalized)

v1 = [0.707107, 0.707107]

v2 = [0.707107, -0.707107]

Λ (Diagonal Matrix)

3.000000	0
0	1.000000

V (Columns are eigenvectors)

0.707107	0.707107
0.707107	-0.707107

Checks

trace(A) = 4.000000; ∑λᵢ ≈ 4.000000

det(A) = 3.000000; ∏λᵢ ≈ 3.000000

Iteration details and notes

3×3 采用未位移 QR 迭代，收敛精度由 ε 与最大迭代 K 控制。可选 shift（占位）与对称优化（占位）在未来版本启用。残差 ||A v - λ v|| 用于衡量每对特征对的数值精度。

Residuals ||A v - λ v||

[0.000e+0, 0.000e+0]

Eigenpairs (vᵢ corresponds to λᵢ)

λ1

3.000000

[0.707107, 0.707107]

λ2

1.000000

[0.707107, -0.707107]

Formulas

2x2 Analytic Solution

Characteristic polynomial: |A - λI| = λ² - (tr A)λ + det A

λ₁,₂ = (tr A ± √((tr A)² - 4 det A)) / 2

Eigenvector v solves (A - λI)v = 0

3x3 Numeric (QR Iteration)

Repeat: Aₖ = Rₖ Qₖ with Aₖ₊₁ = Rₖ Qₖ. Diagonal converges to eigenvalues; columns of accumulated Q approximate eigenvectors.

How to Calculate

1
Choose matrix size
2x2 (analytic) or 3x3 (numeric)
2
Enter row-major entries
a11, a12, ...
3
Submit to compute
Results show eigenvalues and eigenvectors

Important Considerations

Complex Eigenvalues

2x2 matrices with negative discriminant have complex eigenvalues; this tool reports real parts only.

Conditioning

Nearly defective matrices may yield unstable eigenvectors; interpret directions qualitatively.

Normalization

Eigenvectors are normalized to unit length; sign is arbitrary.

Ordering

Vectors are ordered by |λ| descending.

Example Cases

2x2 Example

A = [[2, 1], [1, 2]] → λ = 3, 1; eigenvectors ∥ [1,1]ᵀ and [1,−1]ᵀ

3x3 Example

A = [[3,1,0],[0,2,1],[0,0,1]] → λ ≈ 3, 2, 1; upper triangular shows eigenvalues on diagonal.

Tips

Scale matrix entries to avoid overflow/underflow in numeric steps.
For repeated eigenvalues, any basis of the eigenspace is valid; expect non-unique vectors.
Results are for real inputs; complex routines are not included in this version.

Frequently Asked Questions

What are eigenvalues and eigenvectors?

For a square matrix A, λ is an eigenvalue and v a non-zero eigenvector if A v = λ v. They reveal invariant directions and scaling factors of the linear transformation.

Which matrices are supported?

This tool supports 2x2 (analytic characteristic polynomial) and 3x3 (numeric QR iteration) real matrices.

Can results be complex?

This version reports real eigenvalues/eigenvectors. For 2x2 with negative discriminant, the real part is shown and imaginary parts are omitted.

How accurate is the 3x3 method?

The unshifted QR iteration is suitable for small matrices. Results are typically accurate for well-conditioned inputs. Ill-conditioned matrices may require specialized methods.

How should I format inputs?

Enter row-major matrix entries (a11, a12, …). Use the size toggle for 2x2 or 3x3.

A	col 1	col 2
row 1
row 2