characteristic polynomial
Characteristic Polynomial of a Matrix
Let $A$ be a $n\times n$ matrix over some field $k$. The characteristic polynomial^{} ${p}_{A}(x)$ of $A$ in an indeterminate $x$ is defined by the determinant^{}:
$${p}_{A}(x):=det(AxI)=\left\begin{array}{cccc}\hfill {a}_{11}x\hfill & \hfill {a}_{12}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {a}_{1n}\hfill \\ \hfill {a}_{21}\hfill & \hfill {a}_{22}x\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {a}_{2n}\hfill \\ \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\ddots}\hfill & \hfill \mathrm{\vdots}\hfill \\ \hfill {a}_{n1}\hfill & \hfill {a}_{n2}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill {a}_{nn}x\hfill \end{array}\right$$ 
Remarks

•
The polynomial^{} ${p}_{A}(x)$ is an $n$thdegree polynomial over $k$.

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If $A$ and $B$ are similar matrices^{}, then ${p}_{A}(x)={p}_{B}(x)$, because
${p}_{A}(x)$ $=$ $det(AxI)=det({P}^{1}BPxI)$ $=$ $det({P}^{1}BP{P}^{1}xIP)=det({P}^{1})det(BxI)det(P)$ $=$ $det{(P)}^{1}det(BxI)det(P)=det(BxI)={p}_{B}(x)$ for some invertible matrix $P$.

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The characteristic equation of $A$ is the equation ${p}_{A}(x)=0$, and the solutions to which are the eigenvalues^{} of $A$.
Characteristic Polynomial of a Linear Operator
Now, let $T$ be a linear operator on a vector space^{} $V$ of dimension^{} $$. Let $\alpha $ and $\beta $ be any two ordered bases for $V$. Then we may form the matrices ${[T]}_{\alpha}$ and ${[T]}_{\beta}$. The two matrix representations of $T$ are similar matrices, related by a change of bases matrix. Therefore, by the second remark above, we define the characteristic polynomial of $T$, denoted by ${p}_{T}(x)$, in the indeterminate $x$, by
$${p}_{T}(x):={p}_{{[T]}_{\alpha}}(x).$$ 
The characteristic equation of $T$ is defined accordingly.
Title  characteristic polynomial 

Canonical name  CharacteristicPolynomial 
Date of creation  20130322 12:17:47 
Last modified on  20130322 12:17:47 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  10 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 15A18 
Related topic  Equation 
Defines  characteristic equation 