Review: Eigenvalues
Eigenvalues are the "DNA" of a matrix. They tell us everything about the long-term behavior of a dynamic system.
Introduction
Most vectors change direction when multiplied by a matrix .
Eigenvectors are special: they only change length (stretch/shrink), not direction.
The amount they stretch is the Eigenvalue.
Definition
("lambda") is the eigenvalue.
is the eigenvector (cannot be zero).
How to Find Them
- Solve Characteristic Equation: . This finds .
- For each , solve to find .
Visual: Invariant Direction
Interactive: Matrix Transformation
When the input vector lies on an "Eigenline" (green), the output vector points in the exact same direction (just scaled).
Worked Examples
Example 1: Finding Lambda
Find eigenvalues of .
1. Form Characteristic Eq:
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2. Solve:
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.
.
.
Eigenvalues: .
Example 2: Finding Eigenvector
Find the eigenvector for in Example 1.
1. Plug Lambda into A-LI:
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2. Apply to v:
.
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3. Pick a vector:
Let . Then .
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Example 3: Complex Roots
Find eigenvalues of .
1. Determinant:
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2. Solve:
.
This corresponds to a rotation matrix (rotation by 90 degrees). No real vector points in the same direction after rotation (except 0), so no real eigenvectors.
Practice Quiz
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