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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.