Eigenvalues and Eigenvectors

Eigenvalues and Eigenvectors are important for solving systems of differential equations.

According to Wikipedia, Eigenvalues and Eigenvectors are defined as

In linear algebra, an eigenvector (/ˈaɪɡənˌvɛktər/) or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by ${\displaystyle \lambda }$, is the factor by which the eigenvector is scaled.

You start by finding the eigenvalues, then the eigenvector. These methods only work on square matrices, which should be fine for solving systems of differential equations.

Finding Eigenvalues

For a square matrix $A$, the eigenvalue and eigenvector will make the following equation true:

Where $v$ is the eigenvector and $\lambda$ is the eigenvalue. Square matrices tend to have multiple eigenvalues, and multiple eigenvectors.

To find an eigenvalue, one must solve the characteristic equation for the matrix.

Where $I$ is the identity matrix. Also the | symbol does not mean absolute value but the determinant of the resultant matrix.

Example 1: Find the eigenvalues of the following matrix

The characteristic equation for the above matrix will be the following.

Simplify that to get the following determinant matrix

Now we just have to take the determinant of the left, and that should yield an equation which can be solved for $\lambda$.

The above is a quadratic equation which can be simplified into this.

Now, solve for $\lambda$ using a quadratic equation solving method. You should get $\lambda=-7 \text{ or } 6$.

The eigenvalues of our square matrix $A$ are -7 and 6. With these eigenvalues, we can now find eigenvectors.

Finding Eigenvectors.

To find eigenvectors, we solve the following equation for all the eigenvalues of a matrix.

Remember that $v$ is the eigenvector. We can make this simpler by changing the equation to the following:

Solving the equation for $v$ will find the eigenvector for an eigenvalue. $\lambda_i$ is the identity matrix with $\lambda$ as the coefficent.

Example 1 Continued: Find the eigenvectors for the following matrix

To find the eigenvectors for this matrix, we need the eigenvalues. Since this is the matrix from example 1, we already have the eigenvalues.

Now, we have to plug each value into Equation 3 which was shown above. We will start with $\lambda = 6$.

Start by finding $(A-\lambda_i)$ and set it equal to $B$

So now we have $Bv=0$ because $B=(A-\lambda_i)$. This is a matrix system of equations which can be solved by traditional matrix methods.

This is a simple system of equations, it can be solved using matrix methods or traditional methods like substitution. Once $x$ and $y$ are found, plug that into the following equation to find eigenvector $v$

Solving the following system gets us $x=1$ and $y=4$. So the eigenvector for $\lambda=6$ is

Now, repeat the same process for $\lambda=-7$.

Start by finding $(A-\lambda_i)$ and set it equal to $B$

Do the $Bv=0$ next.

Solve this system of equations to get a value of $v$.

Verifying eigenvalues/vectors

To verify eigenvalues and eigenvectors, just plug the value and the vector into equation 1

If both sides are equal, then the eigenvalue and eigenvector work.

Example 1 continued: Verify the found eigenvalues and eigenvectors.

Start with $\lambda=6$

Matrix multiply the left side and simplify the right side to get the following result:

Because both sides are equal, $\lambda=6$ is a valid eigenvalue.

To verify $\lambda=-7$, just repeat the same process with the corresponding vector for $\lambda=-7$.

Simplifying that will give the following answer.

Again, both sides are equal so $\lambda=7$ and its corresponding eigenvector are valid.