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.