How to Determine the Eigenvalues of a Matrix

How to Determine the Eigenvalues of a Matrix
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  • 0:04 Matrix & Vector Multiplication
  • 0:58 Finding Eigenvalues
  • 2:44 Example 1: 2x2 Matrix
  • 4:37 Example 2: 3x3 Matrix
  • 6:15 Lesson Summary
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Lesson Transcript
Instructor: Damien Howard

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

Together we'll learn how to find the eigenvalues of any square matrix. Once we've got that down we'll practice finding eigenvalues by going through an example with a 2x2 matrix, and one with a 3x3 matrix.

Matrix & Vector Multiplication

Some of the first fundamentals you learn for working with matrices are how to multiply them by scalars, vectors, and other matrices. Multiplying a matrix by a matrix or a scalar gives you another matrix, but multiplying by a vector works a little differently. When you multiply a matrix (A) by a vector (v) you get a new vector (x).


matrix times vector


There's also a special case where, instead of getting a completely new vector, you get a scaled version of the same vector you started with. In other words, a matrix times a vector equals a scalar (lambda) times that same vector.


eigenvalue equation


When this happens we call the scalar (lambda) an eigenvalue of matrix A. How many eigenvalues a matrix has will depend on the size of the matrix. An nxn matrix will have n eigenvalues. In this lesson, we're going learn how to find the eigenvalues of a given matrix.

Finding Eigenvalues

Before getting into examples, we need to find the general solution for finding the eigenvalues of an nxn matrix. To find this, we'll start with our equation from the last section, and rearrange it to get everything on one side of the equals sign, as you can see in the equation on your screen now.


eigenvalue general solution part1


There are a couple of things we need to note here. For one, the zero here is not a scalar, but rather the zero vector. Secondly, we're searching for a solution to the above equation under the condition that v isn't equal to zero. When v equals zero, lambda's value becomes trivial because any scalar or matrix multiplied by a zero vector equals another zero vector.

The next thing we need to do is multiply lambda*v by an identity matrix (I). Multiplying by an identity matrix is like multiplying by one for scalar equations. In other words, it doesn't actually affect the values in our equation, as you can see on screen.


eigenvalue general solution part2


Since both A and lambda*I are multiplied by v, we can factor it out.


eigenvalue general solution part3


When v isn't equal to zero, this equation is true only if the matrix we multiply v by is noninvertible. This means there must not exist a matrix B such that C*B = B*C = I, where C = A - lambda*I in our case. A matrix is noninvertible only when its determinant equals zero, as you can see on your screen right now.


eigenvalue general solution final


When we solve for the determinant, we're going to get a polynomial with eigenvalues as its roots. We call this polynomial the matrix's characteristic polynomial. We can then figure out what the eigenvalues of the matrix are by solving for the roots of the characteristic polynomial.

Example 1: 2x2 Matrix

Let's practice finding eigenvalues by looking at a 2x2 matrix. Earlier we stated that an nxn matrix has n eigenvalues. So a 2x2 matrix should have 2 eigenvalues. For this example, we'll look at the following matrix with 4, 2, 1, and 3.


2x2 matrix


To find the eigenvalues, we're going to use the determinant equation we found in the previous section.


eigenvalue general solution final


First we insert our matrix in for A, and write out the identity matrix. In general, an identity matrix is written as an nxn matrix with ones on the diagonal starting at the top left and zeroes everywhere else, which you can see in the matrices that are appearing on your screen right now. We'll use a 2x2 identity matrix here because we want it to be the same size as A.


2x2 eigenvalues part1


Next we want to simplify everything inside the determinant to get a single matrix.


2x2 eigenvalues part2


Now we're set to solve for the determinant and find the matrix's characteristic polynomial.


2x2 eigenvalues part3


All we have left to do is find the roots of the characteristic polynomial to get our eigenvalues. There are a few different methods you can use to try and find the roots of a second order polynomial, but the only method that always works is using the quadratic equation, which we can see play out here on screen. Let's walk through it step by step:


2x2 eigenvalues solution


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