Polynomials Applied to Linear Operators Examples 1

# Polynomials Applied to Linear Operators Examples 1

Recall from the Polynomials Applied to Linear Operators page that we can apply a polynomial to a linear operator $T \in \mathcal L(V)$ by defining $T^m = \underbrace{T \circ T \circ ... \circ T}_{\mathrm{m \: times}}$ where $m$ is a positive integer. Furthermore, if $T$ is an invertible linear operator then we define $T^{-m} = \underbrace{T^{-1} \circ T^{-1} \circ ... \circ T^{-1}}_{\mathrm{m \: times}}$.

We will now look at some examples regarding polynomials applied to linear operators.

## Example 1

Let $T$ be a linear operator on $V$. Suppose that $T^3 - 6T^2 + 11T - 6I = 0$. Prove that then if $\lambda$ is an eigenvalue of $T$ then $\lambda = 1$ or $\lambda = 2$ or $\lambda = 3$.

Let $\lambda$ be an eigenvalue of $T$. Then for some nonzero vector $u \in V$ we have that $T(u) = \lambda u$. Note that:

(1)
\begin{align} \quad T^2(u) = T(T(u)) = T(\lambda u) = \lambda T(u) = \lambda^2 u \\ \quad T^3(u) = T(T^2(u)) = T(\lambda^2 u) = \lambda^2 T(u) = \lambda^3 u \\ \quad \quad \quad \vdots \quad \quad \quad \\ \quad T^n(u) = T(T^{n-1}(u)) = T(\lambda^{n-1} u) = \lambda^{n-1} T(u) = \lambda^n u \end{align}

Substituting this into $T^3 - 6T^2 + 11T - 6I = 0$ and we have that:

(2)
\begin{align} \quad T^3 - 6T^2 + 11T - 6I = 0 \\ \quad \lambda^3 - 6\lambda^2 + 11 \lambda - 6 \lambda = 0 \\ \quad (\lambda - 1)(\lambda - 2)(\lambda - 3) = 0 \end{align}

Therefore we have that $\lambda = 1$ or $\lambda = 2$ or $\lambda = 3$.

## Example 2

Let $T$ be a linear operator on $V$ such that $T = T^2$. Show that then $V = \mathrm{null} (T) \oplus \mathrm{range} (T)$.