What is Characteristic polynomial? — Mathematics

Definition 3.3 University Mathematics — Year 2 · Chapter 3 — Reduction of Endomorphisms

$\chi_u(X) = \det(X\,\mathrm{id} - u)$ — computed in any basis as $\det(XI_n - A)$ , a monic polynomial of degree $n$ , invariant under similarity (Theorem 2.17). Its roots in $K$ are exactly the eigenvalues ( $\lambda$ eigenvalue $\iff u - \lambda\,\mathrm{id}$ not injective $\iff \chi_u(\lambda) = 0$ ), and

\chi_u(X) = X^n - (\operatorname{tr} u)\, X^{n-1} + \dots + (-1)^n \det u .

The algebraic multiplicity $m_\lambda$ of an eigenvalue is its multiplicity as a root of $\chi_u$ ; the geometric multiplicity is $\dim E_\lambda$ , and $1 \leq \dim E_\lambda \leq m_\lambda$ .

Examples

Example 3.4 (Same $\chi$ , different geometry)

The matrices

\begin{pmatrix}2 & 0\\ 0 & 2\end{pmatrix} \qquad\text{and}\qquad \begin{pmatrix}2 & 1\\ 0 & 2\end{pmatrix}

share the characteristic polynomial $(X - 2)^2$ , the trace, the determinant, the spectrum — yet are not similar: the first has $E_2$ of dimension $2$ (geometric multiplicity $2$ ), the second of dimension $1$ . The characteristic polynomial sees only algebraic multiplicities; the eigenspace dimensions are the finer invariant, and the minimal polynomial arbitrates ( $X - 2$ versus $(X - 2)^2$ ). Moral for all diagonalizability discussions: $\chi$ shortlists the candidates, but kernels cast the votes.

Read in context →

Examples

Related terms