Applying Polynomials to Linear Operators

Definition

1. Let $f=c_0+\cdots +c_k{x}^k$
2. Let $T\in \mathcal{L}(V)$
3. Let $A\in M_{n\times n}(\mathbb{F})$ Then,
4. $f(T)=c_{0}I+c_{1}T+\cdots +c_k{T}^k={\sum }_{i=1}^n{c}_i{T}^i$ (For operators)
5. $f(A)=c_{0}T+c_{1}A+\cdots +c_k{A}^k={\sum }_{i=1}^n{c}_i{A}^i$ (For matrixes)

Example

With $T:{\mathbb{R}}^2\to {\mathbb{R}}^2$ as $T(x_1,x_2)=(x_2,3x_1-x_2)$ Find $f(T)$ where $f(x)=x^2-1$

Soln

1. $f(T)=T^2-I$
2. $=f(T)(x_1,x_2)=(T^2-1)(x_1,x_2)$
3. $=T^2(x_1,x_2)-I(x_1,x_2)$
4. $=T(T(x_1,x_2))-I(x_1,x_2)$
5. $=T(x_2,3x_1-x_2)-(x_1,x_2)$
6. $=(3x_1-x_2,3x_2-(3x_1-x_2))-(x_1,x_2)$
7. $=(2x_1-x_2,3x_2-3x_1)$