Polynomial of an endomorphism

Definition

Let $F$ be a field, let $V$ be a vector space over $F$, and let $T \colon V \to V$ be an endomorphism. Let $p(x) = c_0 + c_1 x + \cdots + c_d x^d$ be a polynomial in $F[x]$. The value of $p$ at $T$ is the endomorphism $$p(T) = c_0 \id_V + c_1 T + c_2 T^2 + \cdots + c_d T^d,$$ where $T^k$ is the composition of $k$ copies of $T$. For an $n \times n$ matrix $A$, $p(A) = c_0 I_n + c_1 A + \cdots + c_d A^d$ in the same way.

Remarks

The map $p \mapsto p(T)$ is a homomorphism of algebras $F[x] \to \End_F(V)$: $(p + q)(T) = p(T) + q(T)$ and $(pq)(T) = p(T) q(T)$. In particular, any two polynomials in $T$ commute. If $T(v) = \lambda v$, then $p(T)(v) = p(\lambda) v$.

Explanations

L1 human

We can put a matrix into a polynomial, because matrices can be added, scaled and multiplied. The constant term becomes a multiple of the identity matrix. For example, if $p(x) = x^2 - 3x + 2$, then $p(A) = A^2 - 3A + 2I$.