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 $A$.