# 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.

Try doing it yourself before looking at the solution below.

Hopefully you got the following:

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:

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 Î», we have the following:

Ax = Î»x

We call Î» an eigenvalue.

So, in our example in the introduction, Î» = 3,

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

A(cy) = Î»cy

cAy = Î»cy

Ay = Î»y

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 = Î»x. First, notice that we can subtract Î»x from both sides, giving us

Ax - Î»x = 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 - Î», 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 - Î»Ix = 0

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

(A - Î»I)x = 0

Since we now have a matrix (A - Î»I) multiplying by a nonzero vector (x) to give us 0, A - Î»I has a determinant of 0. We can use this to find eigenvalues by solving the equation det(A - Î»I) = 0 for Î». Due to the nature of the determinant, det(A - Î»I) 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.

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

(4 - Î»)(1 - Î») - (-1)(2) = 0

meaning that

Î»² - 5Î» + 4 + 2 = 0

Î»² - 5Î» + 6 = 0

(Î» - 3)(Î» - 2) = 0

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