What is Cayley–Hamilton theorem?

The Cayley–Hamilton theorem states that substituting the matrix A for x in polynomial, p(x) = det(xIn – A), results in the zero matrices, such as:

p(A) = 0

It states that a ‘n x n’ matrix A is demolished by its characteristic polynomial det(tI – A), which is monic polynomial of degree n. The powers of A, found by substitution from powers of x, are defined by recurrent matrix multiplication; the constant term of p(x) provides a multiple of the power A0, where power is described as the identity matrix.

The theorem allows An to be articulated as a linear combination of the lower matrix powers of A. If the ring is a field, the Cayley–Hamilton theorem is equal to the declaration that the smallest polynomial of a square matrix divided by its characteristic polynomial.

Example of Cayley-Hamilton Theorem

1.) 1 x 1 Matrices

For 1 x 1 matrix A(a1,1) the characteristic polynomial is given by 

$\begin{array}{l}�(�)=�–�\end{array}$

So, p(A) = (a) – (a1,1)  = 0 is obvious.

2.) 2 x 2 Matrices

Let us look this through an example

$\begin{array}{l}�=\left(\begin{array}{cc}1& 2\\ 3& 4\end{array}\right)\end{array}$

$\begin{array}{l}�(�)=���(�{�}_2-�)=���\left(\begin{array}{cc}�-1& -2\\ -3& �-4\end{array}\right)=(�-1)(�-4)-(-2)(-3)={�}^2-5�-2\end{array}$

The Cayley-Hamilton claims that if, we define

$\begin{array}{l}�(�)={�}^2-5�-2{�}_2\end{array}$

then,

$\begin{array}{l}�(�)={�}^2-5�-2{�}_2=\left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right)\end{array}$

We can verify this result by computation

$\begin{array}{l}{�}^2-5�-2{�}_2=\left(\begin{array}{cc}7& 10\\ 15& 22\end{array}\right)-\left(\begin{array}{cc}5& 10\\ 15& 20\end{array}\right)-\left(\begin{array}{cc}2& 0\\ 0& 2\end{array}\right)=\left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right)\end{array}$

For a generic 2 x 2 matrix,

$\begin{array}{l}�=\left(\begin{array}{cc}�& �\\ �& �\end{array}\right)\end{array}$

The resultant polynomial is given by:

$\begin{array}{l}�(�)={�}^2-(�+�)�+(��-��)\end{array}$

So the Cayley-Hamilton theorem states that

$\begin{array}{l}�(�)={�}^2-(�+�)�+(��-��){�}_2=\left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right)\end{array}$

it is always the case, which is evident by working out on  A2.