LA 101
M04 · L01
Module 4: Eigenvalues
The Eigenvalue Problem
When a matrix acts on a vector and only stretches it — never rotates — that vector is an eigenvector and the stretch factor is its eigenvalue.
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LA 101
M04 · L01
Geometric Intuition
Special Directions
- Most vectors get both rotated and stretched
- Eigenvectors only get stretched — never rotated
- They reveal the transformation's natural axes
- Av = λv: matrix action = simple scaling
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LA 101
M04 · L01
The Definition
The Eigenvalue Equation
A nonzero vector v is an eigenvector of A if multiplying by A only scales it — by eigenvalue λ.
Eigenvalue Equation
A\mathbf{v}=\lambda\mathbf{v}
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LA 101
M04 · L01
From Equation to Determinant
The Characteristic Equation
Rearrange Av = λv → (A − λI)v = 0. For a nonzero solution v to exist, (A − λI) must be singular.
Characteristic Equation
\det(A-\lambda I)=0
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LA 101
M04 · L01
The Characteristic Polynomial
For A = [[3,1],[0,2]]
Polynomial
(3−λ)(2−λ) = λ² − 5λ + 6 = (λ−3)(λ−2)
Eigenvalues
λ₁ = 3 λ₂ = 2
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LA 101
M04 · L01
Step-by-Step Example
Four Steps to Eigenvalues
Step 1
Form A − λI (subtract λ from diagonal)
Step 2
Set det(A − λI) = 0
Step 3
Factor the characteristic polynomial
Step 4
The roots are the eigenvalues
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LA 101
M04 · L01
Counting Eigenvalues
Algebraic Multiplicity
An eigenvalue can repeat as a root of the characteristic polynomial. A degree-n matrix has exactly n eigenvalues counting multiplicity.
n
Eigenvalues
∏λᵢ
= det(A)
∑λᵢ
= tr(A)
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LA 101
M04 · L01
Higher Dimensions
For 3×3 Matrices
- Characteristic polynomial has degree 3
- Up to 3 eigenvalues (counting multiplicity)
- Some may be complex numbers
- Real symmetric matrices → always real eigenvalues
- Triangular matrices → eigenvalues on the diagonal
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LA 101
M04 · L01
Key Properties
What Eigenvalues Tell Us
- det(A) = product of all eigenvalues
- tr(A) = sum of all eigenvalues
- Similar matrices share eigenvalues
- Eigenvalues of triangular matrix are its diagonal entries
- Singular matrix ↔ 0 is an eigenvalue
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LA 101
Up Next
Lesson 1 Complete
M4-L2: Finding Eigenvectors
You've found the eigenvalues. Next: find the eigenvectors — the special directions — and discover what they reveal about the structure of any matrix.
Coming Up
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