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How to Determine the Eigenvectors of a Matrix

Instructor: Damien Howard

Damien has a master's degree in physics and has taught physics lab to college students.

In this lesson, you'll explore the subject of eigenvectors. After learning what an eigenvector is in concept, we'll solidify in your mind how to find them by working through an example problem together.

Row Operations

As children we spent our time in elementary school math classes learning the various ins and outs of the mathematical operations: addition, subtraction, multiplication, and division. In linear algebra you learn these operations all over again. Only this time instead of working with scalars, you're working with vectors and matrices.


scalar vector and matrix examples


If we look at a matrix and a scalar, we can see that the big difference between the two is that the scalar is made of one number and the matrix a bunch of rows and columns of numbers. In addition to learning how to use the four basic operations with matrices, there are an additional three operations that are used on the rows or columns within a single matrix. We call these the row operations, and they are as follows:

1) Any two rows within a matrix can be swapped.


row operation example 1


2) You can multiply each entry in a row by a non-zero scalar.


row operation example 2


3) You can add one row that's been multiplied by a scalar to another row.


row operation example 3


Knowing how to do these row operations is extremely useful for working with matrices. One example of their usefulness comes into play when finding the eigenvectors of a matrix. In the rest of this lesson we'll learn what eigenvectors are, and how to find them for a given matrix.

Eigenvalues and Eigenvectors

In order to understand eigenvectors, we also need to talk about eigenvalues. You cannot find a matrix's eigenvectors without first knowing its eigenvalues.

To see what eigenvalues and eigenvectors, are we start with the following fact. When you multiply a matrix (A) times a vector (v), you get another vector (y) as your answer.


matrix times vector equation


Sometimes the vector you get as an answer is a scaled version of the initial vector.


eigenvalue eigenvector equation 1


When this happens, the scalar (lambda) is an eigenvalue of matrix A, and v is an eigenvector associated with lambda. We find the eigenvectors for a given eigenvalue by solving the following equation for v.


eigenvalue eigenvector equation 2


In this equation, I is an identity matrix the same size as A, and 0 is the zero vector.


identity matrix examples


To find all of a matrix's eigenvectors, you need solve this equation once for each individual eigenvalue.

Example Problem

In order to see exactly how we solve for v, let's work through an example problem.


example problem matrix


The matrix above has eigenvalues (lambda) of 0, -4, and 3. We'll find the eigenvectors associated with lambda = -4.

To find this eigenvector, the first thing we need to do is insert this matrix and eigenvalue into our equation from the previous section and simplify the problem.


example problem step1


The next step in the process is using Gaussian elimination to get our matrix in row echelon form. If you're not already familiar with those two terms, what I've just said probably sounded like gibberish. Let's break down exactly what we'll be doing.

First, what is row echelon form? A matrix is in row echelon form when the following three conditions are met:

1) All zero rows must be at the bottom of the matrix.


row echelon condition 1 example


2) The first entry of all non-zero rows is to the right of the first entry of the row above it.


row echelon condition 2 examples


3) The first entry of every non-zero row is a one.


row echelon condition 3 example


In order to get a matrix in row echelon form, we use Gaussian elimination. This process has two steps. The first is to convert our equation into an augmented matrix.


augmented matrix form


The next step is where those row operations we went over at the beginning of this lesson finally see some use. We use the three row operations on our augmented matrix until we have it in row echelon form. When doing this yourself, the process may require some trial and error until you get the matrix in the right form.


getting row echelon form



getting row echelon from continued


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