San José State University

applet-magic.com
Thayer Watkins
Silicon Valley
USA

 The Cayley*-Hamilton Theorem: Its Nature and Its Proof

Let A be an n×n matrix of real elements. The determinantal equation defining its eigenvalues is

#### det(A−λI) = 0

where I is the n×n identity matrix. This equation in terms of a determinant generates a polynomimal equation p(λ)=0 where p(λ) is called the characteristic polynomial of the matrix. The Cayley-Hamilton Theorem is that if A is substituted for λ in the characteristic polynomial the result is a matrix of zeroes.

## Illustration of the theorem

Let A be the matrix

#### 12 34

Then the matrix A−λI is

#### 1−λ2 34−λ

The determinant of (A−λI) is then

#### (1−λ)(4−λ)−2*3 which reduces to 4 −5λ+λ² −6 and further to −2 −5λ + λ²

This is the characteristic polynomial of the matrix A. The solutions to the polynomical equation

#### −2 −5λ + λ² = 0

are the eigenvalues of the matrix A.

Consider now what results when λ is replaced with A and −2 is replaced with −2I.

A² is the matrix

#### 710 1522

Therefore −2I −5A + A² is

#### −2−5*1+70−5*2+10 0−5*3+15−2−5*4+22 which evaluates to 00 00

Thus, almost mystically, A satisfies its own eigenvalue value equation.

## Quick Proof

The charactistic polynomial equation is equivalent to the determinantal equation

#### det(A−λI) = 0

The expression λI is the diagonal matrix Λ so the determinantal equation is really

#### det(A−Λ) = 0

If A is substituted for Λ in this equation the result is

#### det(A−A) = det(O) = 0

where O is a matrix of zeroes. Certainly A satisfies the determinantal equation.

## The Long Tedious Proof

Let the eigenvalue equation be expressed in the equivalent form

#### det(λI−A) = 0

This removes the ambiguity of the sign of λn in the characteristic equation; it is always 1 for all n.

(To be continued.)

* Cayley is the Irish name usually spelled Kelly. Kelly should be pronounced as Cayley.