Matrix Inversion by the Cayley-Hamilton Theorem

Date: 02/27/98 at 18:34:43
From: DuBois Ford
Subject: Matrix Inversion by way of the Cayley-Hamilton Theorem

I would like to know what the Cayley-Hamilton Theorem is and how it is 
used to find the inverse of a matrix. I have checked many math 
encyclopedias but could only find information on Cayley and Hamilton, 
not the theorem.

Date: 02/28/98 at 10:47:28
From: Doctor Anthony
Subject: Re: Matrix Inversion by way of the Cayley-Hamilton Theorem

The Cayley-Hamilton theorem states that every square matrix A 
satisfies its own characteristic equation.

The characteristic equation is the equation whose roots are the 
eigenvalues of the matrix.  If these terms are unfamiliar, I will 
illustrate with a 2x2 matrix, but the ideas can be generalized to an 
nxn matrix.

    A = [a    b|
        |c    d]

If k is an eigenvalue of the matrix then k is found by solving the 
equation

              |a-k    b | = 0     
              | c    d-k|       (The lefthand side is a determinant)

          (a-k)(d-k) - cb = 0

  ad - ak - dk + k^2 - cb = 0

     k^2 - (a+d)k + ad-bc = 0   This is the characteristic equation 
                                of A.

Then Cayley-Hamilton states that this equation is satisfied by A.

 A^2 - (a+d)A + (ad-bc).I = 0     where I is the identity matrix.

Example:             If A =[1  -1|
                           |2   3]

The characteristic equation of A is 

             k^2 - 4k + 5 = 0  ...........(1)

    so      A^2 - 4A + 5I = 0  ...........(2)

If you are familiar with the ideas of eigenvalues and eigenvectors, 
then if k is an eigenvalue and u an eigenvector, we have

            A.u = k.u     multiply through by A

          A^2.u = A(k.u)

          A^2.u = k.A.u
       
          A^2.u = k^2.u   

Apply these ideas to equations (1) and (2)

Multiply (1) by the vector u, and we have

     k^2.u - 4k.u + 5.u = 0  so replacing k^n.u  by  A^n.u we have

    A^2.u - 4A.u + 5I.u = 0

       (A^2 - 4A + 5I)u = 0  and so

          A^2 - 4A + 5I = 0  and this proves Cayley-Hamilton.


Cayley-Hamilton can be used to find powers of matrices or the 
inverse of a matrix.

For example, if A is the matrix given above, we can write

           A^2 = 4A - 5I
        
               = [4    -4| - [5   0|
                 |8    12]   |0   5]

               = [-1    -4|
                 | 8     7]

To find the inverse of A we write the equation in the form:

              5I = 4A - A^2     now multiply by A^(-1)

         5A^(-1) = 4I - A

         5A^(-1) = [4   0| - [1   -1|
                   |0   4]   |2    3]

                 = [3    1|
                   |-2   1]

          A^(-1) = (1/5)[3    1|
                        |-2   1]

Although I have demonstrated the methods on a 2x2 matrix, the methods 
are clearly valid for any nxn matrix.

-Doctor Anthony,  The Math Forum
 http://mathforum.org/dr.math/