Characteristic Polynomial

linear-algebra

Definition

Characteristic Polynomial

The characteristic polynomial of a square matrix $A\in F^{n\times n}$ is the determinant:

$${\chi }_A(\lambda )=\det (\lambda \cdot I_n-A)$$

Note that the degree of this polynomial is $n$ and the leading coefficient is $1$. Thus, eigenvalues are the rots of ${\chi }_A(\lambda )$. Since a polynomial of degree $n$ (over a field $F$) can have at most $n$ distinct roots, an $n\times n$ square matrix has at most $n$ eigenvalues.