Eigenvectors and Eigenvalues

The directions a transformation refuses to turn.

12 problems · about 24 minutes · N-VM.C.11

What this lesson teaches

The student defines eigenvectors and eigenvalues and identifies them geometrically.

An eigenvector is a direction the transformation does not turn, only stretches.
The eigenvalue is that stretch factor, and a negative one reverses the direction.
A rotation has no real eigenvectors, because every direction turns.

Straightforward practice. Get the method working first.

5 problems

Av = ⟨9, 12⟩ where v = ⟨3, 4⟩. What is the eigenvalue?
Answer 3
Why 3.
What is special about an eigenvector?
Answer The transformation leaves it on its own line, only stretching it.
Why It stays on its own line.
Av = ⟨6, 8⟩ where v = ⟨3, 4⟩. What is the eigenvalue?
Answer 2
Why Double.
Av = ⟨−4, −2⟩ where v = ⟨2, 1⟩. What is the eigenvalue?
Answer -2
Why Reversed and doubled.
A reflection. What is the eigenvalue for a vector perpendicular to the mirror line?
Answer -1
Why It flips to the other side.

The same ideas with more to keep track of.

3 problems

A scaling by 7 in every direction. What is its eigenvalue?
Answer 7
Why Everything stretches by the same factor.
An eigenvalue of 0. Is the matrix singular? 1 yes, 0 no.
Answer 1
Why That direction is crushed.
A 90 degree rotation. How many real eigenvectors does it have?
Answer 0
Why Every direction turns.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Match each transformation to an eigenvalue it has.
Answer The identity → 1; A scaling by 3 → 3; A reflection, perpendicular to the mirror → −1
Why Only the reflection reverses anything.
The identity matrix. What is its eigenvalue?
Answer 1
Why Nothing moves.
The Stretch: Av = ⟨10, 15⟩ where v = ⟨2, 3⟩. What is the eigenvalue?
Answer 5
Why 5.
The Reflection: A reflection across a line. What is the eigenvalue for a vector lying along that mirror line?
Answer 1
Why 1.