Eigenvalues and Eigenvectors

Special directions that are only scaled by a matrix transformation.

An eigenvector is a nonzero vector whose direction is preserved by a matrix transformation. The vector may stretch, shrink, or flip, but it stays on the same line through the origin.

The defining equation is

\[ A\mathbf{v}=\lambda \mathbf{v}. \]

Here \(\mathbf{v}\) is the eigenvector and \(\lambda\) is the eigenvalue, the scale factor.

What you should be able to do

By the end of this lesson, you should be able to:

  • explain the equation \(A\mathbf{v}=\lambda\mathbf{v}\) in words;
  • check whether a proposed vector is an eigenvector;
  • find eigenvalues of simple diagonal or triangular matrices;
  • interpret positive, negative, zero, and large eigenvalues;
  • explain why eigenvectors help with repeated transformations;
  • use SymPy to compute eigenvalues and eigenvectors exactly.

The basic definition

A vector \(\mathbf{v}\) is an eigenvector of \(A\) if:

  1. \(\mathbf{v}\) is not the zero vector;
  2. multiplying by \(A\) produces a scalar multiple of \(\mathbf{v}\).

For example, let

\[ A=\begin{bmatrix}3&0\\0&1\end{bmatrix} \quad \text{and} \quad \mathbf{v}=\begin{bmatrix}1\\0\end{bmatrix}. \]

Then

\[ A\mathbf{v} = \begin{bmatrix}3\\0\end{bmatrix} =3\begin{bmatrix}1\\0\end{bmatrix}. \]

So \(\mathbf{v}\) is an eigenvector and its eigenvalue is \(3\).

WarningThe zero vector is excluded

The zero vector would satisfy \(A\mathbf{0}=\lambda\mathbf{0}\) for every \(\lambda\), which would make the definition meaningless. Eigenvectors must be nonzero because they represent directions.

How to check a proposed eigenvector

To check whether \(\mathbf{v}\) is an eigenvector:

  1. compute \(A\mathbf{v}\);
  2. ask whether the result is a scalar multiple of \(\mathbf{v}\);
  3. if yes, the scalar is the eigenvalue.

Worked example: one vector works, one does not

Let

\[ A=\begin{bmatrix}2&0\\0&5\end{bmatrix}. \]

Check \(\mathbf{v}=\begin{bmatrix}1\\0\end{bmatrix}\):

\[ A\mathbf{v}=\begin{bmatrix}2\\0\end{bmatrix}=2\mathbf{v}. \]

So \(\mathbf{v}\) is an eigenvector with eigenvalue 2.

Now check \(\mathbf{w}=\begin{bmatrix}1\\1\end{bmatrix}\):

\[ A\mathbf{w}=\begin{bmatrix}2\\5\end{bmatrix}. \]

This is not a scalar multiple of \(\begin{bmatrix}1\\1\end{bmatrix}\), because a scalar multiple would have equal coordinates. Therefore \(\mathbf{w}\) is not an eigenvector of this matrix.

Reading eigenvalues from diagonal matrices

For a diagonal matrix

\[ \begin{bmatrix} a&0\\0&d \end{bmatrix}, \]

the coordinate axes are eigenvector directions. The eigenvalues are \(a\) and \(d\). The first coordinate direction is scaled by \(a\) and the second coordinate direction is scaled by \(d\).

What eigenvalues mean

  • \(\lambda=2\): the eigenvector doubles in length.
  • \(\lambda=\frac12\): it shrinks by half.
  • \(\lambda=-1\): it flips direction but keeps the same length.
  • \(\lambda=0\): the direction collapses to the zero vector.
  • \(|\lambda|>1\): repeated multiplication grows along that direction.
  • \(|\lambda|<1\): repeated multiplication shrinks along that direction.

Why eigenvectors matter

If \(A\mathbf{v}=\lambda\mathbf{v}\), then repeated multiplication is easy:

\[ A^2\mathbf{v}=\lambda^2\mathbf{v}, \qquad A^3\mathbf{v}=\lambda^3\mathbf{v}. \]

This idea helps explain long-term behavior in dynamical systems, differential equations, Markov chains, vibrations, computer graphics, and data analysis.

NoteBig idea

Eigenvectors reveal the directions a transformation treats simply. Even if a matrix mixes most vectors, it may have special directions that only scale.

Practice exercises

  1. If \(A\mathbf{v}=7\mathbf{v}\), what is the eigenvalue?
  2. Check whether \(\begin{bmatrix}1\\0\end{bmatrix}\) is an eigenvector of \(\begin{bmatrix}4&0\\0&2\end{bmatrix}\).
  3. Check whether \(\begin{bmatrix}1\\1\end{bmatrix}\) is an eigenvector of \(\begin{bmatrix}4&0\\0&2\end{bmatrix}\).
  4. What are the eigenvalues of \(\begin{bmatrix}3&0\\0&-2\end{bmatrix}\)?
  5. If an eigenvalue has absolute value less than 1, what happens under repeated multiplication along that eigenvector direction?
  1. The eigenvalue is 7.
  2. Yes. The product is \(\begin{bmatrix}4\\0\end{bmatrix}=4\begin{bmatrix}1\\0\end{bmatrix}\).
  3. No. The product is \(\begin{bmatrix}4\\2\end{bmatrix}\), not a scalar multiple of \(\begin{bmatrix}1\\1\end{bmatrix}\).
  4. The eigenvalues are 3 and -2.
  5. The vector shrinks toward the zero vector along that direction.

Subtopic guided practice and checkpoints

Eigenvalue problems ask one central question: does the matrix keep this vector on the same line through the origin?

Lab 1: check whether a vector is an eigenvector

Guess first. If \(A\mathbf{v}=\begin{bmatrix}6\\0\end{bmatrix}\) and \(\mathbf{v}=\begin{bmatrix}3\\0\end{bmatrix}\), is \(\mathbf{v}\) an eigenvector?

Guided exercise.

Let

\[ A=\begin{bmatrix}2&0\\0&5\end{bmatrix}, \qquad \mathbf{v}=\begin{bmatrix}3\\0\end{bmatrix}. \]

Compute

\[ A\mathbf{v}=\begin{bmatrix}6\\0\end{bmatrix}. \]

Now compare this with \(\mathbf{v}\):

\[ \begin{bmatrix}6\\0\end{bmatrix}=2\begin{bmatrix}3\\0\end{bmatrix}. \]

So \(\mathbf{v}\) is an eigenvector and the eigenvalue is \(2\).

Checkpoint. Check whether \(\begin{bmatrix}0\\4\end{bmatrix}\) is an eigenvector of \(\begin{bmatrix}2&0\\0&5\end{bmatrix}\). If yes, find the eigenvalue.

Lab 2: notice when a vector is not on the same line

Guess first. Is \(\begin{bmatrix}2\\5\end{bmatrix}\) a scalar multiple of \(\begin{bmatrix}1\\1\end{bmatrix}\)?

Guided exercise.

A scalar multiple of \(\begin{bmatrix}1\\1\end{bmatrix}\) must have equal coordinates:

\[ c\begin{bmatrix}1\\1\end{bmatrix}=\begin{bmatrix}c\\c\end{bmatrix}. \]

Since \(\begin{bmatrix}2\\5\end{bmatrix}\) has unequal coordinates, it is not a scalar multiple. Therefore if \(A\begin{bmatrix}1\\1\end{bmatrix}= \begin{bmatrix}2\\5\end{bmatrix}\), then \(\begin{bmatrix}1\\1\end{bmatrix}\) is not an eigenvector of \(A\).

Checkpoint. Is \(\begin{bmatrix}-6\\3\end{bmatrix}\) a scalar multiple of \(\begin{bmatrix}-2\\1\end{bmatrix}\)? If yes, what is the scalar?

Lab 3: read eigenvalues from diagonal matrices

Guess first. What are the eigenvalues of \(\begin{bmatrix}-3&0\\0&4\end{bmatrix}\)?

Guided exercise.

A diagonal matrix scales coordinate directions separately:

\[ \begin{bmatrix}-3&0\\0&4\end{bmatrix} \begin{bmatrix}1\\0\end{bmatrix} = \begin{bmatrix}-3\\0\end{bmatrix} =-3\begin{bmatrix}1\\0\end{bmatrix}, \]

and

\[ \begin{bmatrix}-3&0\\0&4\end{bmatrix} \begin{bmatrix}0\\1\end{bmatrix} = \begin{bmatrix}0\\4\end{bmatrix} =4\begin{bmatrix}0\\1\end{bmatrix}. \]

So the eigenvalues are \(-3\) and \(4\).

Checkpoint. Find the eigenvalues and coordinate-axis eigenvectors of \(\begin{bmatrix}7&0\\0&\frac12\end{bmatrix}\).

Lab 4: interpret the sign and size of an eigenvalue

Guess first. If \(\lambda=-2\), does the vector shrink, grow, flip, or both grow and flip?

Guided exercise.

If \(A\mathbf{v}=-2\mathbf{v}\), then applying \(A\) makes the vector twice as long and points it in the opposite direction. The negative sign flips direction. The absolute value \(2\) tells the stretch factor.

Checkpoint. Describe what happens along an eigenvector when \(\lambda=0.25\), \(\lambda=-1\), and \(\lambda=3\).

Lab 5: use eigenvalues for repeated multiplication

Guess first. If \(A\mathbf{v}=3\mathbf{v}\), what is \(A^4\mathbf{v}\)?

Guided exercise.

Each application of \(A\) multiplies the vector by \(3\):

\[ A\mathbf{v}=3\mathbf{v}, \quad A^2\mathbf{v}=3^2\mathbf{v}, \quad A^3\mathbf{v}=3^3\mathbf{v}, \quad A^4\mathbf{v}=3^4\mathbf{v}=81\mathbf{v}. \]

This is why eigenvectors are useful for understanding long-term behavior.

Checkpoint. If \(A\mathbf{v}=\frac12\mathbf{v}\), what is \(A^5\mathbf{v}\)?

  1. \(\begin{bmatrix}0\\4\end{bmatrix}\) is an eigenvector because the product is \(\begin{bmatrix}0\\20\end{bmatrix}=5\begin{bmatrix}0\\4\end{bmatrix}\). The eigenvalue is \(5\).
  2. Yes. \(\begin{bmatrix}-6\\3\end{bmatrix}=3\begin{bmatrix}-2\\1\end{bmatrix}\).
  3. The eigenvalues are \(7\) with eigenvector \(\begin{bmatrix}1\\0\end{bmatrix}\) and \(\frac12\) with eigenvector \(\begin{bmatrix}0\\1\end{bmatrix}\).
  4. \(0.25\) shrinks toward zero without flipping. \(-1\) flips direction and keeps the same length. \(3\) grows by a factor of 3 without flipping.
  5. \(A^5\mathbf{v}=(\frac12)^5\mathbf{v}=\frac{1}{32}\mathbf{v}\).

Eigenvector guessing game

Use this checkpoint to identify eigenvalues, eigenvectors, and repeated behavior.

Using this lesson with edumath and SymPy

SymPy is very useful for eigenvalue work because it can return exact symbolic answers. In a PyScript/Pyodide page, the same commands can be used after loading SymPy.

from edumath.linear_algebra import eigenvalues_2x2, validate_eigenpair

A = [[3, 0], [0, 1]]
v = [1, 0]

print(eigenvalues_2x2(A))
print(validate_eigenpair(A, v, 3).message)
((3+0j), (1+0j))
Correct.
import sympy as sp

A = sp.Matrix([[3, 0], [0, 1]])
print(A.eigenvals())
print(A.eigenvects())
{3: 1, 1: 1}
[(1, 1, [Matrix([
[0],
[1]])]), (3, 1, [Matrix([
[1],
[0]])])]