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(A-xI)=\left|\begin{matrix}a_{11}-x&a_{12}&\cdots&a_{1n}\\ a_{21}&a_{22}-x&\cdots&a_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ a_{n1}&a_{n2}&\cdots&a_{nn}-x\end{matrix}\right|

Remarks

•

The polynomial $p_{A}(x)$ is an $n$ th-degree polynomial over $k$ .

•

If $A$ and $B$ are similar matrices, then $p_{A}(x)=p_{B}(x)$ , because

$\displaystyle p_{A}(x)$	$\displaystyle=$	$\displaystyle\det(A-xI)=\det(P^{-1}BP-xI)$
	$\displaystyle=$	$\displaystyle\det(P^{-1}BP-P^{-1}xIP)=\det(P^{-1})\det(B-xI)\det(P)$
	$\displaystyle=$	$\displaystyle\det(P)^{-1}\det(B-xI)\det(P)=\det(B-xI)=p_{B}(x)$

for some invertible matrix $P$ .

•

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 $n<\infty$ . 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	2013-03-22 12:17:47
Last modified on	2013-03-22 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