Cayley-Hamilton Theorem
For a
square matrix
, identity matrix
, and zero matrix
, the following holds:
This is called the
Cayley-Hamilton Theorem.
[Note] For
, the Cayley-Hamilton theorem gives
. However, matrix
is not unique in satisfying the equation
. For example,
also satisfies
.
Example.
Given the matrix
satisfies the following equation.
Find the value of
for the constants
and
.
Solution
Since
(
is a real number), by the Cayley-Hamilton theorem,