Eigenvalues: Definition, Properties & Examples

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  • 0:03 Definition of Eigenvalues
  • 0:22 Some Matrix Information
  • 2:51 Finding Eigenvalues
  • 7:10 Eigenvalue Properties
  • 7:56 Lesson Summary
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Lesson Transcript
Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

Two important features of a matrix are the eigenvector and the eigenvalue. In this lesson, we'll explore the definition and properties of eigenvalues through examples.

Definition of Eigenvalues

When looking at an acorn, do you see a tree? Probably not. Yet an acorn contains the genetic information for a tree. An acorn is like an eigenvalue, which condenses the information in a matrix. Let's use some examples to explore the definition and the properties of eigenvalues.

Some Matrix Information

To keep things manageable, the examples we'll use will be matrices having a maximum of 2 rows and 2 columns (but the concepts can be extended for larger matrices). You can see our sample matrix:


A sample 2x2 matrix called A


Let's start with the main diagonal, which starts at the top left and goes to the bottom right in the matrix. The numbers on the main diagonal of A are -5 and 4. The trace is the sum of the numbers on the main diagonal. The trace of A is -5 + 4 = -1. So far, so good?

In general, if our 2x2 matrix had letters instead of numbers, we could write some formulas for the determinant, a useful number that is calculated from the entries in a matrix. A generic matrix could look like this one:


A generic 2x2 matrix


To find the determinant, we multiply a times d and subtract the product of b times c. As a formula, the determinant = ad - bc. This works for 2x2 matrices. The determinant of A is -5(4) - (-7)2 = -20 + 14 = -6.

An mx1 matrix (i.e., m rows and 1 column) is called a column vector. In general, multiplying a 2x2 matrix by a 2x1 column vector looks like:


Multiplying a 2x2 matrix by a 2x1 column vector


To get some practice, let's multiply A by the vector (1 -1) :


A times one of its eigenvectors


This vector was special! When we multiply this vector by the matrix A, we get the vector back, although it is scaled by a number. This vector is called an eigenvector of A and the scaling number is an eigenvalue associated with the eigenvector. Remember how an acorn is condensed tree information? Well, when we have two distinct eigenvalues and their associated eigenvectors for a 2x2 matrix, we can reconstruct the matrix. How cool is that!

Finding Eigenvalues

Remember what happened when we multiplied the matrix A with an eigenvector of A? The eigenvector was unchanged except for a scaling factor. If we had multiplied any other vector by A, the vector would have changed. It's only these special eigenvectors that remain the same. An equation summarizing this is Av = λv where λ is the eigenvalue associated with the eigenvector v.

To find the eigenvalues, we take the determinant of A - λI, set this result to zero, and solve the eigenvalues λ. I represents the identity matrix, which has 1 along the main diagonal and 0 everywhere else. Before using this determinant equal to zero idea, you might be wondering where this comes from.

Okay, λv is the same as λIv. Then, Av = λv is the same as Av = λIv. So, Av - λIv = 0 or (A - λI)v = 0. Great! We want to find eigenvectors v and eigenvalues λ. This last equation is true if v = 0 but having a vector equal to zero is not interesting. We want solutions for v not equal to 0. Now, A - λI is a matrix. What if A - λI had an inverse? Then,


matrix times its inverse is the identity matrix


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