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Cayley-Hamilton Theorem

Module by: Thanos Antoulas, JP Slavinsky

Summary: 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 AA,

xAλ=detλI-A xA λ λ I A (1)
this theorem says that xAA=0 xA A 0 . Looking at an expanded form of this definition, let us say that xAλ=λn+α n - 1 λn-1+…+α1λ+α0 xA λ λ n α n - 1 λ n 1 … α1 λ α0

Cayley-Hamilton tells us that we can insert the matrix AA in place of the eigenvalue variable λλ and that the result of this sum will be 00: An+α n - 1 An-1+…+α1A+α0I=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.

An= - α n - 1 An-1-…-α1A-α0I A n - α n - 1 A n 1 … α1 A α0 I (2)

Example 1

Take the following matrix and its characteristic polynomial. A=2111 A 21 11 xAλ=λ2-3λ+1 xA λ λ 2 3 λ 1 Plugging AA into the characteristic polynomial, we can find an expression for A2 A 2 in terms of AA and the identity matrix: A2-3A+I=0 A 2 3 A I 0

equation of characteristic polynomial

A2=3A-I A 2 3 A I (3)

To compute A2 A 2 , we could actually perform the matrix multiplication, as below: A2=21112111=5332 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 AA by 33 and then subtracting 11 from the elements on the diagonal. A2=6333-I=5332 A 2 63 33 I 53 32

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