Polinômio característico

Em álgebra linear, o polinómio característico de uma matriz $A$ $n\times n$ ou de um operador linear $A\in L(V,V)$ em um espaço vetorial $V$ de dimensão finita $n$ com base $C$ é o polinómio:

$p_{A}(x)=\det[xI-A]_{C}$

em que $\det$ é o determinante e $I$ é a matriz identidade $n\times n$ (ou o operador identidade). Este é um polinómio mônico de grau $n$, ou seja, o coeficiente do termo de maior grau é $1$. Os autovalores de $A$ são as raízes de seu polinómio característico.

O polinómio minimal de um operador linear A em L(V, V) é o polinómio mônico m_A(x) de menor grau tal que $m_{A}(A)(v)=0$, $\forall v\in V$.

Motivação

Uma matriz quadrada "A" é singular se, e somente se, 0 é um autovalor de A. Esta é, aliás, a principal técnica para descobrir se uma matriz é singular:

$det\left(A-\lambda I\right)=0$.

Para uma matriz de dimensão nXn, o lado esquerdo desta equação é um polinómio de grau n na variável λ, denominado polinômio característico de A.

Exemplo

Seja $A$ uma matriz de dimensão $2\times 2$ dada por:

$A={\begin{bmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{bmatrix}}$

Então, seu polinômio característico é:

${\begin{aligned}p_{A}(\lambda )&=\det \left(A-\lambda I\right)\\&=\det \left({\begin{bmatrix}a_{11}&a_{12}\\a_{21}&a_{22}\end{bmatrix}}-\lambda {\begin{bmatrix}1&0\\0&1\end{bmatrix}}\right)\\&=\det \left({\begin{bmatrix}a_{11}-\lambda &a_{12}\\a_{21}&a_{22}-\lambda \end{bmatrix}}\right)\\&=\left[\left(a_{11}-\lambda \right)\left(a_{22}-\lambda \right)\right]-\left[a_{12}a_{21}\right]\\&=\lambda ^{2}-\left(a_{11}+a_{22}\right)\lambda +\left(a_{11}a_{22}-a_{12}a_{21}\right)\\&=\lambda ^{2}-{\text{tr}}(A)\lambda +\det(A)\end{aligned}}$

onde, ${\text{tr}}(A)$ é o traço de $A$.

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