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# Cayley-Hamilton theorem

In linear algebra, the Cayley-Hamilton theorem (named after the mathematicians Arthur Cayley[?] and William Hamilton) states that every square matrix over a commutative ring, e.g. over the real or complex field, satisfies its own characteristic equation. This means the following: if A is the given square matrix and

$p(t)=\det(A-tI)$

is its characteristic polynomial (a polynomial in the variable t), then replacing t by the matrix A results in the zero matrix:

$p(A)=0.$

Consider for example the matrix

$A = \begin{pmatrix}1&2\\ 3&4\end{pmatrix}$. The characteristic polynomial is given by
$p(t)=\det\begin{pmatrix}1-t&2\\ 3&4-t\end{pmatrix}=(1-t)(4-t)-(2)(3)=t^2-5t-2.$ The Cayley-Hamilton theorem then claims that
$A^2-5A-2I_2=0$
which one can quickly verify in this case.

The theorem is an important tool in calculating eigenvectors.

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