Eigenvalues & Eigenvectors — Interactive Guide

1What Are Eigenvalues & Eigenvectors?

A matrix A acts on vectors — it multiplies them and usually changes their direction. But certain special vectors come out pointing in the same (or opposite) direction as they went in. These are eigenvectors.

A · x = λ · x

🔵 Eigenvector (x)

A non-zero vector that does not change direction when multiplied by A. It may be stretched, shrunk, or flipped — but it stays on the same line.

🔴 Eigenvalue (λ)

The scalar multiplier. It tells you how much the eigenvector is stretched (λ > 1), shrunk (0 < λ < 1), flipped (λ < 0), or zeroed out (λ = 0).

Note: λ can be positive, negative, zero, or even complex. x must be a non-zero vector.

Special Case: λ = 0

If λ = 0, then A·x = 0. This means x is in the null space of A, and A is a singular matrix (no inverse).

A is singular ↔ λ = 0 is an eigenvalue

2How to Find Eigenvalues & Eigenvectors

Start from A·x = λ·x and rearrange:

A·x − λ·x = 0 → (A − λI)·x = 0

For a non-zero x to exist, the matrix (A − λI) must be singular. A matrix is singular when its determinant is zero:

det(A − λI) = 0
← Characteristic Equation

Form (A − λI) Subtract λ from every entry on the main diagonal of A. Leave all other entries unchanged.

Set det(A − λI) = 0 Compute the determinant and set it equal to zero. This gives the characteristic polynomial in λ.

Solve for λ (the Eigenvalues) Find the roots of the characteristic polynomial. An n×n matrix has exactly n eigenvalues (counting repeats).

Find Eigenvectors for each λ For each eigenvalue λᵢ, solve (A − λᵢI)·x = 0. Use elimination to find the null space. The result is the eigenvector(s).

2×2 Shortcut: The characteristic equation is always λ² − (trace)·λ + (det) = 0. Use this to skip the determinant expansion!

3Vector Transformation Visualizer

Watch how matrix A transforms vectors. Eigenvectors (🔵 blue) stay on the same line — they only get scaled by λ. Other vectors (🔴 red) change direction.

Matrix A:

4Interactive 2×2 Eigenvalue Calculator

Enter your 2×2 matrix A:

Results will appear here…

5Worked Examples

Symmetric Matrix

Permutation

Projection

Rotation (Complex)

Triangular (Repeated)

Matrix: A = [[3,1],[1,3]] (Symmetric matrix)

A = | 3 1 |
| 1 3 |

Step 1 & 2: Characteristic Equation ▼

det(A − λI) = (3−λ)(3−λ) − 1·1
= λ² − 6λ + 8 = 0

Shortcut check: trace = 3+3 = 6 ✓ | det = 9−1 = 8 ✓

Step 3: Eigenvalues ▼

(λ−4)(λ−2) = 0
λ₁ = 4, λ₂ = 2

Step 4: Eigenvectors ▼

For λ₁ = 4:

A − 4I = | −1 1 | → x₁ = | 1 |
| 1 −1 | | 1 |

For λ₂ = 2:

A − 2I = | 1 1 | → x₂ = | −1 |
| 1 1 | | 1 |

✓ Both eigenvectors are perpendicular: (1)(−1) + (1)(1) = 0. This always happens for symmetric matrices!

Eigenvalue	Eigenvector	Meaning
λ₁ = 4	x₁ = [1, 1]ᵀ	Stretched by factor 4
λ₂ = 2	x₂ = [−1, 1]ᵀ	Stretched by factor 2

Matrix: A = [[0,1],[1,0]] (Swaps the two components of any vector)

A = | 0 1 |
| 1 0 |

trace = 0, det = −1

λ² − 0·λ + (−1) = 0 → λ² − 1 = 0
λ₁ = 1, λ₂ = −1

Eigenvalue	Eigenvector	Intuition
λ₁ = 1	x₁ = [1, 1]ᵀ	Swapping two equal entries → same vector
λ₂ = −1	x₂ = [−1, 1]ᵀ	Swapping opposite entries → flips sign

Verify: sum of λ = 1 + (−1) = 0 = trace ✓ | product = 1×(−1) = −1 = det ✓

Projection Matrix P onto a plane (3D) — no computation needed, just geometry!

Case 1 — Vector in the plane:

P·x = x → λ = 1

Projection of a vector already in the plane = itself. Unchanged.

Case 2 — Vector ⊥ to the plane:

P·x = 0 → λ = 0

Projection of a perpendicular vector = zero. Collapses to origin.

Eigenvalue	Which eigenvectors?
λ = 1	All vectors lying inside the projection plane
λ = 0	All vectors perpendicular to the plane (null space of P)

Key insight: We can read eigenvalues purely from the geometry — no determinant calculation needed for special matrices!

Rotation Matrix Q — rotates every vector by 90°. No real vector stays on the same line after 90° rotation!

Q = | 0 −1 |
| 1 0 |

trace = 0, det = +1

λ² + 1 = 0
λ₁ = i, λ₂ = −i

Complex eigenvalues! A perfectly real matrix can have complex (imaginary) eigenvalues. This happens when the geometry has no real fixed direction (like rotation).

Matrix Type	Eigenvalue Type
Symmetric (Aᵀ = A)	Always real
Anti-symmetric (Aᵀ = −A)	Pure imaginary
Rotation matrix	Complex (a ± bi)

Upper Triangular Matrix — a case with a repeated eigenvalue.

A = | 3 1 |
| 0 3 |

Key fact: For triangular matrices, eigenvalues = diagonal entries!

det(A − λI) = (3−λ)² = 0
λ₁ = λ₂ = 3 (repeated!)

Now find eigenvectors for λ = 3:

A − 3I = | 0 1 | → only solution: x = | 1 |
| 0 0 | | 0 |

Warning: Shortage of eigenvectors! This 2×2 matrix has only ONE independent eigenvector instead of two. This is called a defective or degenerate matrix. Repeated eigenvalues may (but don't always) cause this problem.

6Key Properties

Trace = Sum of Eigenvalues

a₁₁ + a₂₂ + … + aₙₙ = λ₁ + λ₂ + … + λₙ

The sum of the diagonal entries of A always equals the sum of all eigenvalues.

Determinant = Product of Eigenvalues

det(A) = λ₁ · λ₂ · … · λₙ

This means if any eigenvalue is 0, the determinant is 0 → A is singular.

Adding c·I Shifts All Eigenvalues by c

If A·x = λ·x, then (A + cI)·x = (λ + c)·x

The eigenvectors stay exactly the same. Only the eigenvalues shift by c.

Try it: Shift eigenvalues by adding c·I

A = [[3,1],[1,3]] has λ₁=4, λ₂=2. Now add c·I:

c = 0

λ₁ = 4 + 0 = 4 | λ₂ = 2 + 0 = 2

Eigenvectors stay: [1,1]ᵀ and [−1,1]ᵀ — unchanged!

⚠ Eigenvalues Are NOT Linear

eigenvalues(A+B) ≠ eigenvalues(A) + eigenvalues(B)

This only works when B = c·I. In general, A and B have different eigenvectors, so you cannot simply add their eigenvalues. You must solve the eigenvalue problem from scratch for A+B.

Quick Reference

Property	Formula / Fact
Eigenvalue equation	A·x = λ·x
Characteristic equation	det(A − λI) = 0
2×2 shortcut	λ² − (trace)λ + (det) = 0
Triangular matrix	Eigenvalues = diagonal entries
Symmetric matrix	Real λ, perpendicular eigenvectors
Singular matrix	λ = 0 is always an eigenvalue
Count	n×n matrix → exactly n eigenvalues
Identity shift	(A + cI) has same eigenvectors, λ shifts by c

7Common Pitfalls

🌀 Complex Eigenvalues

Rotation and anti-symmetric matrices can have complex eigenvalues even though the matrix entries are real. Always check if the discriminant (trace² − 4·det) is negative.

🔁 Repeated Eigenvalues

A repeated eigenvalue doesn't automatically mean a shortage of eigenvectors — but it can. Triangular matrices with repeated diagonal entries are the classic example.

➕ Non-Linearity

Eigenvalues don't add or multiply across matrix sums/products (unless one matrix is a multiple of I). Always re-solve for compound matrices.

📐 Eigenvector ≠ Unique

Any scalar multiple of an eigenvector is also an eigenvector. When we "find the eigenvector," we're really finding a basis for the eigenspace (a whole line of vectors).

8Self-Check Quiz