PlanetMath: characteristic polynomial

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (200)
Orphanage
Unclass'd
Unproven (490)
Corrections (6)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

characteristic polynomial

(Definition)

Characteristic Polynomial of a Matrix

Let be a $n \times n$ matrix over some field . The characteristic polynomial of in an indeterminate is defined by the determinant:

$\displaystyle p_A(x):=\det(A-x I) = \left\vert\begin{matrix} a_{11}-x & a_{12} ... ...ddots & \vdots \ a_{n1} & a_{n2} & \cdots & a_{nn}-x \end{matrix} \right\vert$

Remarks

The polynomial is an th-degree polynomial over .
If and are similar matrices, then , 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 .
The characteristic equation of is the equation , and the solutions to which are the eigenvalues of .

Characteristic Polynomial of a Linear Operator

Now, let be a linear operator on a vector space of dimension $n<\infty$ . Let $\alpha$ and $\beta$ be any two ordered bases for . Then we may form the matrices $[T]_{\alpha}$ and $[T]_{\beta}$ . The two matrix representations of are similar matrices, related by a change of bases matrix. Therefore, by the second remark above, we define the characteristic polynomial of , denoted by , in the indeterminate , by

$\displaystyle p_T(x):=p_{[T]_{\alpha}}(x).$

The characteristic equation of

is defined accordingly.

"characteristic polynomial" is owned by CWoo. [ full author list (2) | owner history (1) ]

(view preamble)