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### 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/
```
Associated Topics:
College Linear Algebra

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