Implementation of Cayley-Hamilton's Theorem in MATLAB

In linear algebra, there is a fundamental theorem caled Cayley-Hamilton’s Theorem that is used to establish a relation between a square matrix and its characteristic equation. Read this article to learn the implementation of the Cayley-Hamilton’s theorem in MATLAB programming. Let's start with an overview of Cayley-Hamilton’s theorem.

What is Cayley-Hamilton’s Theorem?

In linear algebra, the Cayley-Hamilton’s theorem is a fundamental theorem that is used to establish a relationship between a square matrix and its characteristic equation. In other words, according Cayley-Hamilton’s theorem, a square matrix must satisfy its own characteristic equation.

Explanation

Consider a square matrix A of the order of "n". If $\mathrm{p(\lambda)}$ is the characteristic polynomial of the matrix A, then it is given by,

$$\mathrm{p(\lambda) \: = \: \lvert \lambda I \: − \: A \rvert}$$

Where,

• "I" is the identity matrix of the order of "n".

• $\mathrm{"\lambda"}$ is a real constant.

The roots of this polynomial $\mathrm{p(\lambda)}$ are called the "eigenvalues" of the square matrix A.

As per Cayley-Hamilton’s theorem, if the polynomial $\mathrm{p(\lambda)}$ is the characteristic polynomial of a square matrix "A". Then, if we substitute $\mathrm{\lambda \: = \: A}$ in $\mathrm{p(\lambda)}$, it gives a zero matrix i.e.,

$$\mathrm{p(A) \: = \: 0}$$

Therefore, if we evaluate the characteristic polynomial of a square matrix A at the matrix A itself, then it results in a zero matrix. Hence, the Cayley-Hamilton’s theorem satisfies its own characteristic polynomial.

The Cayley-Hamilton’s theorem plays a vital role in various field of engineering like control theory, physics, signal processing, and more. This is all about basics of Cayley-Hamilton’s theorem. Let us now learn its implementation in MATLAB.

Implementation of Cayley-Hamilton’s Theorem in MATLAB

We can implement the Cayley-Hamilton’s theorem in MATLAB. The two simplest methods of implementing the Cayley-Hamilton’s theorem are explained below.

Method (1) – Cayley-Hamilton’s Theorem by Defining a Function

function CH(A)
   % Check whether the input matrix is a square matrix
   [m, n] = size(A);
   if m ~= n
      error('Input matrix is not a square matrix.');
   end
    
   % Defining characteristic polynomial of input square matrix
   p = poly(A);
    
   % Cayley-Hamilton theorem calculation
   r = polyvalm(p, A);
    
   % Display the result
   disp('Cayley-Hamilton Theorem:');
   disp(r);
    
   % Verify the Cayley-Hamilton Theorem
   if isequal(round(r), zeros(size(A)))
      disp('Cayley-Hamilton theorem is verified.');
   else
      disp('Cayley-Hamilton theorem verification failed.');
   end
end

Save this function in the current workspace in your MATLAB environment. Then, open the MATLAB command window, and call the "CH" function to compute the Caley-Hamilton’s theorem.

Example 1

% Create a square matrix
A = [10, 20; 30, 40];

% Calculate Cayley-Hamilton’s theorem
CH(A);

Output

It will produce the following output −

Cayley-Hamilton Theorem:
    0     0
    0     0
Cayley-Hamilton theorem is verified.

Example 2

% Create a square matrix
A = [10, 20 30; 30, 40, 50];

% Calculate Cayley-Hamilton’s theorem
CH(A);

Output

It will produce the following output −

Error using CH
Input matrix is not a square matrix.

Let us now discuss another method of implementing the Cayley-Hamilton’s theorem in MATLAB.

Method (2) – Cayley-Hamilton Theorem Calculating using "for" Loop

The following example program demonstrates the implementation of Cayley-Hamilton’s theorem in MATLAB.

Example 3

% MATLAB program to Implement Cayley-Hamilton’s theorem 
% Input a square matrix
A = [10, 20; 30, 40];

% Check if the input matrix is a square matrix
[m, n] = size(A);
if m ~= n
   error('Input matrix is not a square matrix.');
end

% Store the size of the input matrix in a variable
mat_size = size(A);

% Characteristic polynomial calculation
p = poly(A);

% Create a zero matrix of the same size as A
r = zeros(mat_size);

% Calculate Cayley-Hamilton theorem using a loop
for i = 1 : (mat_size(1) + 1)
   r = r + p(i) * (A^(mat_size(1)+1 - i));
end

% Display the result
disp("Cayley-Hamilton’s Theorem:");
disp(round(r));

% Verify Cayley-Hamilton theorem
if isequal(round(r), zeros(mat_size))
   disp("Cayley-Hamilton theorem is verified.")
else
   disp("Cayley-Hamilton theorem verification is failed.")
end

Output

It will produce the following output −

Cayley-Hamilton’s Theorem:
    0     0
    0     0
Cayley-Hamilton theorem is verified.

If we replace the input matrix "A" from a non-square matrix given below −

A = [10, 20, 30; 30, 40, 50];

Then, the output will be −

Input matrix is not a square matrix.

Conclusion

In conclusion, the Cayley-Hamilton’s theorem is a powerful result in linear algebra that gives a relationship between a square matrix and its characteristic equation or polynomial. We can implement the Cayley-Hamilton’s theorem in MATLAB. In this article, I explained the two most simple methods of implementing the Cayley-Hamilton’s theorem in MATLAB with the help of examples. You can try these MATLAB codes with different square matrices to see the results.