## Proof of the Cayley - Hamilton Theorem

The Cayley Hamilton Theorem says that if a matrix
$A$
gives rise to a polynomial equation
$P( \lambda )=\lambda^n + c_{n-1}\lambda^{n-1}+... +c_1 \lambda +c_0=0$
(1)
then
$P(A)=A^n+ c_{n-1}A^{n-1}+... +c_1 A+c_0I=0$
(2).
It can be easily shown that these two equations are equivalent by diagonalizing the matrix
$A$
using the matrix
$P$
of eigenvectors and the relationship
$D=P^{-1}AP=$
where
$D$
is the diagonal matrix of eigenvalues, then
$D^n=(P^{-1}AP)(P^{-1}AP)...(P^{-1}AP)=P^{-1}A^nP$
so multiplying (2) on the left by
$P^{-1}$
and on the right by
$P$
we can write
\begin{aligned} P^{-1}(P( A))P &= P^{-1}A^nP + c_{n-1}P^{-1}A^{n-1}P+...+c_1 P^{-1}AP +P^{-1}c_0P \\ &= D^n+c_{n-1}D^{n-1}+... +c_1D+c_0I =0 \end{aligned}

Now the equations (1) can be read off line by l;ine. (2) can be derived by the reverse process.