Endomorphism (of a vector space)

Definition

An endomorphism is a linear map $T:V \to V$ where $V$ is a vector space over a field $\mathbb{F}$ .

Properties

The endomorphisms of a vector space form a non-commutative ring, and we can define polynomials of endomorphisms.

By picking a basis for $V$ we can represent $T$ as a square matrix $A$ .

If $V$ is finite-dimensional, and $A$ is a matrix representing $T$ , then the characteristic polynomial is defined to be $\chi_T(\lambda) = \det(A-\lambda I)$ . The roots of the characteristic polynomial are the eigenvalues of the map.

If $V$ is finite-dimensional, there exists a uniquely defined minimal polynomial $m_T(x)$ such that that $m_T(T) = 0$ ; equivalently the minimal polynomial such that $m_T(A) = 0$ (regardless of the chosen basis). For any polynomial $f \in \mathbb{F}[t]$ with $f(T) = 0$ , we have $m_T | f$ . The Cayley-Hamilton theorem states that the $\chi_T(T)=0$ and therefore $m_T | \chi_T$ .

We can consider a vector space with a chosen endomorphism as a module over $\mathbb{F}[t]$ where the action of the polynomial $t$ is defined by $t.v \mapsto T(v)$ ; more generally $f(t).v \mapsto f(T)(v)$ . If a minimal polynomial exists (which is true when $\dim V < \infty$ ), then this is a torsion module.

Endomorphism (of a vector space)

Definition

Properties

See also