# Definition:Characteristic Polynomial

## Definition

A **characteristic polynomial** is a member of the class of polynomials which in some way allows one to sum up a number of characteristics of a particular mathematical object.

## Disambiguation

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**Characteristic Polynomial** may refer to:

### Matrix

Let $R$ be a commutative ring with unity.

Let $\mathbf A$ be a square matrix over $R$ of order $n > 0$.

Let $\mathbf I_n$ be the $n \times n$ identity matrix.

Let $R \sqbrk x$ be the polynomial ring in one variable over $R$.

The **characteristic polynomial of $\mathbf A$** is the determinant of the **characteristic matrix of $\mathbf A$ over $R \sqbrk x$**:

- $\map {p_{\mathbf A} } x = \map \det {\mathbf I_n x - \mathbf A}$

### Linear Operator

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.

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

### Field Extension

Let $K$ be a field.

Let $L / K$ be a finite field extension of $K$.

Then by Vector Space on Field Extension is Vector Space, $L$ is naturally a vector space over $K$.

Let $\alpha \in L$, and $\theta_\alpha$ be the linear operator:

- $\theta_\alpha: L \to L : \beta \mapsto \alpha \beta$

The **characteristic polynomial** of $\alpha$ with respect to the extension $L / K$ is:

- $\det \sqbrk {X I_L - \theta_\alpha}$

where:

- $\det$ denotes the determinant of a linear operator
- $X$ is an indeterminate
- $I_L$ is the identity mapping on $L$.

### Element of Algebra

Let $A$ be a commutative ring with unity.

Let $B$ be an algebra over $A$ such that $B$ is a finite-dimensional free module over $A$.

Let $b \in B$.

The **characteristic polynomial** of $b$ is the characteristic polynomial of the regular representation $\lambda_b : B \to B$ over $A$.