= Characteristic polynomial from traces of powers
{title2=$p_k=\operatorname{tr}(A^k)$}
For a complex $n\times n$ <matrix> $A$, $\operatorname{tr}(A^k)$ is the sum of the $k$th powers of its <eigenvalues>, counted with <algebraic multiplicity>. This does not require <diagonalizability>: a similar <upper triangular matrix> has the <eigenvalues> on its diagonal, and taking powers raises those diagonal entries to their corresponding powers. <Newton's identities> therefore recover its <characteristic polynomial> from the first $n$ such <traces>.
For the <Jordan block> $A=\begin{pmatrix}2&1\\0&2\end{pmatrix}$, $\operatorname{tr}A=4$ and $\operatorname{tr}(A^2)=8$. Hence $e_1=4$ and $e_2=(4^2-8)/2=4$, giving \b[$\det(zI-A)=z^2-4z+4=(z-2)^2$]. The repeated <eigenvalue> is recovered even though this <matrix> is not <diagonalizable>.<Rational Univariate Representation (Lecture 12) by Jan Verschelde>[§5, pp. 4–5]{ref}
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