Eigenvalues & Eigenvectors: Definition, Equation & Examples

Eigenvalues & Eigenvectors: Definition, Equation & Examples
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  • 0:04 An Example Matrix
  • 0:42 Eigenvalues & Eigenvectors
  • 1:37 Finding Eigenvalues &…
  • 4:58 Representative Eigenvectors
  • 6:10 Lesson Summary
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Lesson Transcript
Instructor: Michael Gundlach
Every square matrix has special values called eigenvalues. These special eigenvalues and their corresponding eigenvectors are frequently used when applying linear algebra to other areas of mathematics.

An Example Matrix

Every square matrix has special values called eigenvalues. What are these? Well, let's start by doing the following matrix multiplication problem where we're multiplying a square matrix by a vector.


Matrix multiplication


Try doing it yourself before looking at the solution below.

Hopefully you got the following:


Matrix multiplication


What do you notice about the product? If you look closely, you'll notice that it's 3 times the original vector. In fact, we could write our solution like this:


Eigenvalue problem


This tells us that 3 is an eigenvalue, with the original vector in the multiplication problem being an eigenvector.

Eigenvalues & Eigenvectors

An eigenvector of a square matrix A is a nonzero vector x such that for some number λ, we have the following:

Ax = λx

We call λ an eigenvalue.

So, in our example in the introduction, λ = 3,


Eigenvector


Notice that if x = cy, where c is some number, then

A(cy) = λcy

cAy = λcy

Ay = λy

Therefore, every constant multiple of an eigenvector is an eigenvector, meaning there are an infinite number of eigenvectors, while, as we'll find out later, there are a finite amount of eigenvalues. Each eigenvalue will have its own set of eigenvectors.

Finding Eigenvalues & Eigenvectors

We need to find the eigenvalues to find the eigenvectors. To do so, we're going to manipulate the equation Ax = λx. First, notice that we can subtract λx from both sides, giving us

Ax - λx = 0

where 0 represents the zero vector, or the column vector made up of only zeros.

Next, we want to factor out x on the left side of the equation, but to do so, we need to take care of two important details. First, notice that if we factor x out without being careful, we get A - λ, which is problematic. This is a problem since we can't subtract a number from a matrix; we can only subtract a matrix of the same size.

Therefore, we're going to rewrite x as Ix. We can do this since I is the identity matrix; multiplying against it does nothing. This gives us

Ax - λIx = 0

The second important detail we need to take into account is that the order of multiplication matters with matrices. Therefore,

(A - λI)x = 0

Since we now have a matrix (A - λI) multiplying by a nonzero vector (x) to give us 0, A - λI has a determinant of 0. We can use this to find eigenvalues by solving the equation det(A - λI) = 0 for λ. Due to the nature of the determinant, det(A - λI) will always be an nth degree polynomial when A is an n by n matrix, meaning there will be n solutions if we count the ones that are complex numbers. Therefore, an n by n matrix has n eigenvalues. As an example, we're going to find the eigenvalues of the following 2 by 2 matrix.


Matrix


Finding an eigenvalue


Using the determinant formula for 2 by 2 matrices, we get that

(4 - λ)(1 - λ) - (-1)(2) = 0

meaning that

λ² - 5λ + 4 + 2 = 0

λ² - 5λ + 6 = 0

(λ - 3)(λ - 2) = 0

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