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 $A \sqbrk x$ be the polynomial ring in one variable over $A$.
Let $I_M$ denote the identity mapping on $M$.
Let $M \otimes_A A \sqbrk x$ be the extension of scalars of $M$ to $A \sqbrk x$.
The characteristic polynomial of $\phi$ is the determinant of the linear operator $x I_M - \phi$ on $M \otimes_A A \sqbrk x$.
Also see
- Results about characteristic polynomials of linear operators can be found here.