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
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 Cayley-Hamilton theorem then claims that
-
which one can quickly verify in this case.
As a result of this, the Cayley-Hamilton theorem allows us to calculate powers of matrices more simply than by direct multiplication.
Taking the result above
-
Then, for example, to calculate
A4, observe
-
-
The theorem is also an important tool in calculating
eigenvectors.