# Matrix/Field/Characteristic polynomial/Definition

Characteristic polynomial

For an -matrix with entries in a field , the polynomial

is called the * characteristic polynomial*
of .

Characteristic polynomial

For an ${}n\times n$-matrix ${}M$ with entries in a field ${}K$, the polynomial

- ${}\chi _{M}:=\det {\left(X\cdot E_{n}-M\right)}\,$

is called the * characteristic polynomial*
of ${}M$.