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:

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

  $p_A(x)$$=$$det(A-xI)=det(P^{-1}BP-xI)$

  $=$$det(P^{-1}BP-P^{-1}xIP)=det(P^{-1})det(B-xI)det(P)$

  $=$$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 . 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.