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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