The characteristic polynomial of a square matrix is a polynomial that is invariant under matrix similarity and has the eigenvalues as roots.

Let $A$ be a square $n\times n$ matrix. Its characteristic polynomial $p(\lambda)$ is $\det(\lambda I - A)$. Some authors prefer to define it as $\det(A-\lambda I)$ instead; by multiplicity of the determinant, these definitions agree for $n$ even and agree up to a sign for $n$ odd.

The characteristic polynomial contains a lot of information about $A$. Some properties include:

  • Its constant term is equal to $\det(A)$. In particular, if $p(0)\neq 0$, $A$ is invertible.
  • The coefficient of $\lambda^{n-1}$ is the trace of $A$
  • If $p$ has $n$ distinct roots, $A$ is diagonalizable (this condition is sufficient but not necessary)

A powerful result known as the Cayley-Hamilton theorem states that every matrix satisfies its characteristic polynomial; that is, $p(A)=0$. This is deeper than it appears at first: note that this does not follow by putting $\lambda=A$, as that is an abuse of notation.

A related concept is the minimial polynomial of $A$. Put simply, the characteristic polynomial allows for repeated roots and the minimal polynomial does not.