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.
