Most vectors get knocked off their own line by a transformation. A few do not: they end up longer or shorter, but still pointing along the same line.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The definition
An eigenvector satisfies Av = λv. The transformation acts on it as though it were mere multiplication by a number.
The eigenvalue
That number λ is the eigenvalue: the factor by which the eigenvector is stretched. It may be negative, which reverses the direction.
Zero eigenvalues
An eigenvalue of zero means that direction is crushed to nothing. That happens exactly when the matrix is singular.
Reading them off a picture
A reflection has an eigenvalue of 1 along the mirror line and −1 perpendicular to it. A scaling has every direction as an eigenvector.
When there are none
A genuine rotation turns every direction, so it has no real eigenvectors at all. The eigenvalues turn out to be complex.
Directions the transformation refuses to turn
An eigenvector is a nonzero vector whose direction is unchanged by the transformation; it is only scaled. The scaling factor is the eigenvalue.
Why those directions matter
Along an eigenvector the transformation is just multiplication by a number. Finding enough eigenvectors turns a complicated matrix into a simple one, which is the whole motivation.
Reading them off a picture
A reflection has eigenvectors along the mirror line, with eigenvalue 1, and perpendicular to it, with eigenvalue −1. A rotation in the plane has no real eigenvectors at all, since every direction turns.
Where they appear
Vibration modes, principal components in statistics, stability of dynamical systems, and the ranking algorithm behind early web search. Eigenvectors identify a system's natural directions.
Step 2: Try It Yourself
Tap and try it out.
- i-hat lands on(3, 1)
- j-hat lands on(1, 3)
- Determinant8
- v lands on(4, 4)
- Eigenvalues4 and 2
The dashed lines are the eigen-directions. A vector on one of those lines still points the same way after the transformation, only longer or shorter, and the eigenvalue is that stretch factor.
The shaded parallelogram is the image of the unit square, and its area is 8. That is exactly what the determinant measures.
Step 3: Watch an Example
One step at a time.
Watch Selin Verify an Eigenvector
Selin checks whether ⟨1, 1⟩ is an eigenvector of the matrix with rows [3, 1] and [1, 3].
- Step 1
She multiplies: the first entry is 3(1) + 1(1) = 4.
Step 4: Your Turn
Practice makes it stick.
The Stretch
Problem 1 of 2
Av = ⟨10, 15⟩ where v = ⟨2, 3⟩. What is the eigenvalue?
The Reflection
Problem 2 of 2
A reflection across a line. What is the eigenvalue for a vector lying along that mirror line?
Find the Fixed Directions
1 of 8
Av = ⟨6, 8⟩ where v = ⟨3, 4⟩. What is the eigenvalue?
2 of 8
Av = ⟨−4, −2⟩ where v = ⟨2, 1⟩. What is the eigenvalue?
3 of 8
A reflection. What is the eigenvalue for a vector perpendicular to the mirror line?
4 of 8
A scaling by 7 in every direction. What is its eigenvalue?
5 of 8
An eigenvalue of 0. Is the matrix singular? 1 yes, 0 no.
6 of 8
A 90 degree rotation. How many real eigenvectors does it have?
7 of 8
Match each transformation to an eigenvalue it has.
Tap a card on the left to start.
8 of 8
The identity matrix. What is its eigenvalue?
Step 5: Quick Check
Show what you know.
Question 1 of 2
Av = ⟨9, 12⟩ where v = ⟨3, 4⟩. What is the eigenvalue?
Question 2 of 2
What is special about an eigenvector?
What You Learned
- 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.