Cayley-Hamilton Theorem
Given
(1) | |||
(2) |
then
(3) |
where is the identity matrix. Cayley verified this identity for and 3 and postulated that it was true for all . For , direct verification gives
(4) | |||
(5) | |||
(6) |
(7) | |||
(8) | |||
(9) | |||
(10) | |||
(11) |
so
(12) |
The Cayley-Hamilton theorem states that an matrix is annihilated by its characteristic polynomial , which is monic of degree .