# Eigenvalues and Eigenvectors Examples 4

Recall from the Eigenvalues and Eigenvectors page that the number $\lambda \in \mathbb{F}$ is said to be an eigenvalue of the linear operator $T \in \mathcal L (V)$ if $T(u) = \lambda u$ for some nonzero vector $u \in V$. The nonzero vectors $u$ such that $T(u) = \lambda u$ are called eigenvectors corresponding to the eigenvalue $\lambda$.

We will now look at some examples regarding eigenvalues of linear operators and eigenvectors corresponding to eigenvalues.

## Example 1

**Let $T \in \mathcal ( \wp (\mathbb{R} )$ be defined by $T(p(x)) = p'(x)$ for all $p(x) \in \wp (\mathbb{R})$. Find all eigenvalues of $T$ and their corresponding eigenvectors.**

We want to find numbers $\lambda \in \mathbb{R}$ such that:

(1)

For $p(x) = a_0 + a_1x + a_2x^2 + … + a_nx^n$ we have that:

(2)

We thus have that:

(3)

From the last equation we see that $\lambda = 0$ or $a_n = 0$.

If $\lambda = 0$, then all of the equations above are satisfied, so $\lambda_1 = 0$ is an eigenvalue of $T$. Furthermore, we will have that $a_1 = a_2 = … = a_n = 0$, and $a_0 \in \mathbb{R}$. Therefore the set of polynomials $\{ p(x) = a_0 : a_0 \in \mathbb{R} \: , a_0 \neq 0 \}$ are the corresponding eigenvectors to $\lambda_1 = 0$.

Now if $a_n = 0$, then this implies that $a_{n-1} = … = a_2 = a_1 = a_0 = 0$ which means that $p(x)$ is a zero vector.

Therefore the only eigenvalue is $\lambda = 0$.

## Example 2

**Let $T \in \mathcal L ( \wp_3 (\mathbb{R}))$ be defined by $T(p(x)) = xp'(x)$. Find all eigenvalues of $T$ and their corresponding eigenvectors.**

Let $p(x) = a_0 + a_1x + a_2x^2 + a_3x^3$. We want to then find $\lambda \in \mathbb{F}$ such that:

(4)

From above, we obtain the following system of equations:

(5)

The first equation above implies that either $\lambda = 0$ or $a_0 = 0$.

If $\lambda = 0$ then all of the equations are satisfied and so $\lambda_1 = 0$ is an eigenvalue. We also have that $a_1 = a_2 = a_3 = 0$, and so the polynomials in $\{ p(x) = a_0 : a_0 \in \mathbb{R} \: a_0 \neq 0 \}$ are the corresponding eigenvalues.

If $a_0 = 0$, then the system of equations from above reduces down to:

(6)

Now suppose that $a_1 \neq 0$. Then $\lambda = 1$. This then implies that $a_2 = a_3 = 0$, and so $\lambda_2 = 1$ is an eigenvalue of $T$ and $\{ p(x) = a_1x : a_1 \in \mathbb{R} \: a_1 \neq 0 \}$ is the set of corresponding eigenvectors.

Now suppose that $a_1 = 0$. Then the system of equations from above reduces down to:

(7)

Now suppose that $a_2 \neq 0$. Then $\lambda = 2$. Then this implies that $a_3 = 0$, and so $\lambda_3 = 2$ is an eigenvalue of $T$, and $\{ p(x) = a_2x^2 : a_2 \in \mathbb{R} \: a_2 \neq 0 \}$ is the set of corresponding eigenvectors to the eigenvalue $\lambda_3 = 2$.

Lastly suppose that $a_2 = 0$. Then the system of equations from above reduces down to:

(8)

Suppose that $a_3 \neq 0$. Then $\lambda = 3$. Thus we have that $\lambda_4 = 3$ is an eigenvalue of $T$, and $\{ p(x) = a_3x^3 : a_3 \in \mathbb{R} \: a_3 = 0 \}$ is the set of corresponding eigenvectors to the eigenvalue $\lambda_4 = 3$.

Note that if $a_3 = 0$ then since $a_0 = a_1 = a_2 = a_3 = 0$, we have that $p(x) = 0$ which not a nonzero vector.