Category:Characteristic Polynomials of Linear Operators

This category contains results about Characteristic Polynomials of Linear Operators.
Definitions specific to this category can be found in Definitions/Characteristic Polynomials of Linear Operators.

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 $M \otimes_A A \sqbrk x$.