# 33 Eigenvalues

Eigenvalues are very important and have many applications. (A couple of which are studying population growth and solving certain types of differential equations seen in engineering and science [16].) They have grown from the need to solve the eigenvalue problem.

Larson and Falvo give the following statement for the eigenvalue problem: “If is an matrix, do there exist nonzero matrices x such that is a scalar multiple of x?” [16]. Note that the eigenvalue problem only applies to matrices.

The scalar is denoted as and is called the eigenvalue, and x is the eigenvector [16]. So, what we want to know is what eigenvalues and eigenvectors satisfy the equation

which can be rewritten as

To find the eigenvalues, we solve the characteristic equation

for . This is because the homogeneous equation has a nonzero solution only if the determinant of is equal to zero [16]. In other words, we want to find such that has a nonzero solution. After finding , we use the equation

where 0 is the zero vector, to find the corresponding eigenvector. Note that we do not let x equal zero because is just the trivial solution [16]. Let’s now look at an example that I completed for Elementary Linear Algebra that comes from Larson and Falvo [16].

Example 67
Find the the eigenvalues and the corresponding eigenvectors for

Solution
Using the characteristic equation and solving for , we have

This implies that equals -3 or 6. To find the corresponding eigenvectors, we have to solve for x when and when For ,

Using elementary row operations, we get
The solutions to the equation

which can be written as

are matrices that satisfy

by matrix multiplication. If we let , where is a real number, we have eigenvectors of the form . For ,

Using elementary row operations, we get
The solutions to the equation

which can be written as

are matrices that satisfy

by matrix multiplication. If we let , where is a real number, we have eigenvectors of the form .
Therefore, the eigenvalues are

with corresponding eigenvectors that are nonzero scalar multiples of

The eigenvalue problem is not the only one that exists in Linear Algebra. There is another one known as the diagonalization problem that we will discuss in the following section.

#### Diagonalization

The diagonalization problem goes like this: “For a square matrix , does there exist an invertible matrix such that is diagonal?” [16]. A diagonal matrix is a square matrix where all entries above and below the main diagonal are zeros. So, the only thing differentiating one diagonal matrix from another is what entries are located along the main diagonal. We will see that the diagonalization of a matrix , if it can be done, will result in a diagonal matrix with the eigenvalues of along the main diagonal [16].

Let’s look at the formal definition of a diagonalizable matrix as given by Larson and Falvo [16].

Definition VI.12
An matrix is diagonalizable if is similar to a diagonal matrix. That is, is diagonalizable if there exists an invertible matrix such that is a diagonal matrix.

Once the eigenvalues and corresponding eigenvectors of a matrix are found, it is not difficult to determine whether is diagonalizable. It is also not difficult to find the diagonal matrix if is diagonalizable. We simply have to follow the steps that are given below by Larson and Falvo [16].

Let be an matrix.

1. Find linearly independent eigenvectors for with corresponding eigenvalues . If linearly independent eigenvectors do not exist, is not diagonalizable.
2. If has linearly independent eigenvectors, let be the matrix whose columns are these eigenvectors.
3. The diagonal matrix will have the eigenvalues on its main diagonal (and zeros elsewhere). Additionally, the order of the eigenvectors used to form will correspond to the order in which the eigenvalues appear on the main diagonal of .

Let’s now work through an example demonstrating these steps. This is a problem I wrote for this paper.

Example 68
Find a matrix such that is diagonal where

and find the matrix

Solution
By example 67, the eigenvalues of are -3 and 6 with corresponding eigenvectors

If one of the two eigenvectors could be written as a linear combination of the other, then we would have to find a scalar that could be multiplied by one of the eigenvectors to get the other. It is easy to see that neither can be written as a linear combination of the other. Thus, they are linearly independent, and is diagonalizable. This concludes the first step.

The second step says to let be the matrix that whose columns are the eigenvectors. So, we have that

By step 3, we have that the diagonal matrix has the eigenvalues along the main diagonal appearing in the same order as their corresponding eigenvectors appear in . So,

The solution to the diagonalization problem for is that there does exist an invertible matrix such that is diagonal, and