Chemistry - Cayley-Hamilton theorem

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In linear algebra, the Cayley-Hamilton theorem (named after the mathematicians Arthur Cayley and William Hamilton) states that every square matrix over the real or complex field, satisfies its own characteristic equation. This means the following: if A is the given square nxn matrix and I_n is the nxn identity matrix, then the characteristic polynomial of A is defined as:

$p(t)=\det(A-tI_n)\,$

where "det" is the determinant function. The Cayley-Hamilton theorem states that replacing t by the matrix A in the characteristic polynomial results in the zero matrix:

$p(A)=0.\,$

Indeed, the Cayley-Hamilton theorem holds for square matrices over commutative rings as well.

An important corollary of the Cayley-Hamilton theorem is that the minimal polynomial of a given matrix is a divisor of its characteristic polynomial. This is very useful in finding the Jordan form of a matrix.

Example

Consider for example the matrix

$A = \begin{pmatrix}1&2\\ 3&4\end{pmatrix}$ .

The characteristic polynomial is given by

$p(t)=\det\begin{pmatrix}1-t&2\\ 3&4-t\end{pmatrix}=(1-t)(4-t)-(2)(3)=t^2-5t-2.$

The Cayley-Hamilton theorem then claims that

A 2 - 5 A - 2 I 2 = 0

which one can quickly verify in this case.

As a result of this, the Cayley-Hamilton theorem allows us to calculate powers of matrices more simply than by direct multiplication.

Taking the result above

A 2 - 5 A - 2 I 2 = 0

A 2 = 5 A + 2 I 2 .

Then, for example, to calculate A⁴, observe

A 3 = (5 A + 2 I 2) A = 5 A 2 + 2 A = 5(5 A + 2 I 2) + 2 A = 27 A + 10 I 2

A 4 = A 3 A = (27 A + 10 I 2) A = 27 A 2 + 10 A = 27(5 A + 2 I 2) + 10 A

A 4 = 145 A + 54 I 2 .

The theorem is also an important tool in calculating eigenvectors.

Categories: Linear algebra | Theorems