Eigenvalue/Eigenvector Definition

Linear Algebra for Data Science in R

Eric Eager

Data Scientist at Pro Football Focus

Definition

For a matrix $A$, the scalar $\lambda$ is an eigenvalue of $A$, with associated eigenvector $\vec{v} \neq \vec{0}$ if the following equation is true: $$A\vec{v} = \lambda \vec{v}.$$

In other words:

The matrix multiplication $A\vec{v}$, a matrix-vector operation, produces the same vector as $\lambda \vec{v}$ a scalar multiplication acting on a vector.

This matrix does not have to be like the matrices in the last lecture.

Example

print(A)

     [,1] [,2]
[1,]    2    3
[2,]    0    1

Notice that $\lambda = 2$ is an eigenvalue of $A$ with eigenvector $\vec{v} = (1, 0)^T$:

A%*%c(1,0)

     [,1]
[1,]    2
[2,]    0

2*c(1, 0)

2 0

Geometric Motivation

Example, cont'd

Notice that $\lambda = 2$ is an eigenvalue of $A$ with eigenvector $\vec{v} = (1, 0)^T$ and $\vec{v} = (4, 0)^T$:

A%*%c(1,0)

     [,1]
[1,]    2
[2,]    0

2*c(1, 0)

2 0

A%*%c(4,0)

     [,1]
[1,]    8
[2,]    0

2*c(4, 0)

8 0

Let's practice!

Linear Algebra for Data Science in R