# Definition:Characteristic Polynomial of Linear Operator

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## Definition

Let $A$ be a commutative ring with unity.

Let $M$ be a free module over $A$ of finite rank $n > 0$.

Let $\phi : M \to M$ be a linear operator.

### Definition 1

The **characteristic polynomial** of $\phi$ is the characteristic polynomial of the relative matrix of $\phi$ with respect to a basis of $M$.

### Definition 2

Let $A[x]$ be the polynomial ring in one variable over $A$.

Let $\operatorname{id}$ be the identity mapping on $M$.

Let $M \otimes_A A[x]$ be the extension of scalars of $M$ to $A[x]$.

The **characteristic polynomial** of $\phi$ is the determinant of the linear operator $x\operatorname{id} - \phi$ on $M \otimes_A A[x]$.