CayleyHamilton theorem
In
linear algebra, the
\'CayleyHamilton 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
is its
characteristic polynomial (a
polynomial in the variable
t), then replacing
t by the matrix
A results in the zero matrix:
Consider for example the matrix
 .
The characteristic polynomial is given by

The CayleyHamilton theorem then claims that

which one can quickly verify in this case.
As a result of this, the CayleyHamilton theorem allows us to calculate powers of matrices more simply than by direct multiplication.
Taking the result above

Then, for example, to calculate
A^{4}, observe


The theorem is also an important tool in calculating
eigenvectors.