Cayley-Hamilton Theorem

In linear algebra, the Cayley–Hamilton theorem (termed after the mathematicians Arthur Cayley and William Rowan Hamilton) says that every square matrix over a commutative ring (for instance the real or complex field) satisfies its own typical equation.

If A is a provided as n×n matrix and In is the n×n identity matrix, then the distinctive polynomial of A is articulated as

Where the determinant operation is det and the scalar element of the base ring is λ. As the entries of the matrix are (linear or constant) polynomials in λ, the determinant is also an n-th order monic polynomial in λ. The Cayley–Hamilton theorem says that substituting the matrix A for λ in this polynomial results in the zero matrix,

The powers of A, found by substitution from powers of λ, are defined by recurrent matrix multiplication; the constant term of p(λ) provides a multiple of the power A0, which power is described as the identity matrix. The theorem allows An to be articulated as a linear combination of the lower matrix powers of A. If the ring is a field, the Cayley–Hamilton theorem is equal to the declaration that the smallest polynomial of a square matrix divided by its characteristic polynomial.

Examples of Cayley-Hamilton

1 x 1 Matrices

For 1 x 1 matrix A $(a_{1,1})$ the characteristic polynomial is given by $p(\lambda )=\lambda – A$ and so

p(A) = (a) – $a_{1,1}$ = 0 is obvious.

2 x 2 Matrices

Let us look this through an example

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

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

The Cayley-Hamilton claims that if, we define

p(X) = $X^{2}-5X-2I_{2}$

then,

p(A) = $A^{2}-5A-2I_{2}$$\begin{pmatrix} 0 & 0\\ 0 & 0 \end{pmatrix}$

We can verify this result by computation

$A^{2}-5A-2I_{2}$$\begin{pmatrix} 7 & 10\\ 15 & 22 \end{pmatrix}-\begin{pmatrix} 5 & 10\\ 15 & 20 \end{pmatrix}-\begin{pmatrix} 2 & 0\\ 0 & 2 \end{pmatrix}=\begin{pmatrix} 0 & 0\\ 0 & 0 \end{pmatrix}$

For a generic 2 x 2 matrix,

$A=\begin{pmatrix} a & b\\ c & d \end{pmatrix}$

the resultant polynomial is given by $A=\begin{pmatrix} a & b\\ c & d \end{pmatrix}$, so the Cayley-Hamilton theorem states that

$p(A)=A^{2}-(a+d)A+(ad-bc)I_{2}=\begin{pmatrix} 0 & 0\\ 0 & 0 \end{pmatrix}$

it is always the case, which is evident by working out on $A^{2}$ .

The theorem was first proved in 1853 in terms of inverses of linear functions of quaternions, a non-commutative ring, by Hamilton. This parallels to the special case of certain real 4 × 4 real or 2 × 2 complex matrices. The theorem holds for broad quaternionic matrices. Cayley in 1858 said it for 3 × 3 and smaller matrices, but only published a proof for the 2 × 2 case. The general case was first verified by Frobenius in 1878.