Cayley-Hamilton Theorm 2.13 2001/04/02 2002/10/24 Thanos Antoulas aca@rice.edu John Paul Slavinsky jps@alumni.rice.edu Roy Ha rha@rice.edu Elizabeth Chan lizychan@rice.edu Thanos Antoulas aca@rice.edu John Paul Slavinsky jps@alumni.rice.edu characteristic polynomial cayley hamilton The Cayley-Hamilton Theorm. In vivid color. The Cayley-Hamilton Theorem states that every matrix satisfies its own characteristic polynomial. Given the following definition of the characteristic polynomial of A, xA λ λ I A this theorem says that xA A 0 . Looking at an expanded form of this definition, let us say that xA λ λ n α n - 1 λ n 1 … α1 λ α0 Cayley-Hamilton tells us that we can insert the matrix A in place of the eigenvalue variable λ and that the result of this sum will be 0: A n α n - 1 A n 1 … α1 A α0 I 0 One important conclusion to be drawn from this theorem is the fact that a matrix taken to a certain power can always be expressed in terms of sums of lower powers of that matrix. A n - α n - 1 A n 1 … α1 A α0 I Take the following matrix and its characteristic polynomial. A 21 11 xA λ λ 2 3 λ 1 Plugging A into the characteristic polynomial, we can find an expression for A 2 in terms of A and the identity matrix: A 2 3 A I 0 equation of characteristic polynomial A 2 3 A I To compute A 2 , we could actually perform the matrix multiplication, as below: A 2 21 11 21 11 53 32 Or taking equation of characteristic polynomial to heart, we can compute (with fewer operations) by scaling the elements of A by 3 and then subtracting 1 from the elements on the diagonal. A 2 63 33 I 53 32