# Linear Algebra : Eigenvalues and Eigenvectors of Symmetric Matrices

## Example Questions

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### Example Question #1 : Eigenvalues And Eigenvectors Of Symmetric Matrices

Find the Eigen Values for Matrix .

There are no eigen values

Explanation:

The first step into solving for eigenvalues, is adding in a  along the main diagonal.

Now the next step to take the determinant.

Now lets FOIL, and solve for .

Now lets use the quadratic equation to solve for .

So our eigen values are

### Example Question #2 : Eigenvalues And Eigenvectors Of Symmetric Matrices

Find the eigenvalues and set of mutually orthogonal

eigenvectors for the following matrix.

No eigenvalues or eigenvectors exist

Explanation:

In this problem, we will get three eigen values and eigen vectors since it's a symmetric matrix.

To find the eigenvalues, we need to minus lambda along the main diagonal and then take the determinant, then solve for lambda.

This can be factored to

Thus our eigenvalues are at

Now we need to substitute  into or matrix in order to find the eigenvectors.

For .

Now we need to get the matrix into reduced echelon form.

This can be reduced to

This is in equation form is , which can be rewritten as . In vector form it looks like,

We need to take the dot product and set it equal to zero, and pick a value for , and .

Let , and .

Now we pick another value for , and  so that the result is zero. The easiest ones to pick are , and .

So the orthogonal vectors for  are , and .

Now we need to get the last eigenvector for .

After row reducing, the matrix looks like

So our equations are then

, and , which can be rewritten as .

Then eigenvectors take this form, . This will be orthogonal to our other vectors, no matter what value of , we pick. For convenience, let's pick , then our eigenvector is.

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