Proof of the Cayley - Hamilton Theorem

The Cayley Hamilton Theorem says that if a matrix
\[A \]
 gives rise to a polynomial equation
\[P( \lambda )=\lambda^n + c_{n-1}\lambda^{n-1}+... +c_1 \lambda +c_0=0 \]
\[P(A)=A^n+ c_{n-1}A^{n-1}+... +c_1 A+c_0I=0\]
It can be easily shown that these two equations are equivalent by diagonalizing the matrix
\[A \]
  using the matrix
  of eigenvectors and the relationship
  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
\[\begin{equation} \begin{aligned} P^{-1}(P( A))P &= P^{-1}A^nP + c_{n-1}P^{-1}A^{n-1}P+...+c_1 P^{-1}AP +P^{-1}c_0P \\ &= D^n+c_{n-1}D^{n-1}+... +c_1D+c_0I =0 \end{aligned} \end{equation}\]

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

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