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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

<math>p(t)=\det(A-tI)</math>

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

<math>p(A)=0.</math>

Consider for example the matrix

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

The theorem is an important tool in calculating eigenvectors.



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