Cayley-Hamilton theorem

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 In  is the nxn identity matrix, then the characteristic polynomial of A is defined as:

[itex]p(t)=\det(A-tI_n)\,[itex]

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:

[itex]p(A)=0.\,[itex]

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

[itex]A = \begin{pmatrix}1&2\\

3&4\end{pmatrix}[itex]. The characteristic polynomial is given by

[itex]p_A(t)=\begin{vmatrix}1-t&-2\\

-3&4-t\end{vmatrix}=(1-t)(4-t)-(-2)(-3)=t^2-5t-2.[itex] The Cayley-Hamilton theorem then claims that

[itex]A^2-5A-2I_2=0[itex]

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

[itex]A^2-5A-2I_2=0[itex]
[itex]A^2=5A+2I_2.[itex]

Then, for example, to calculate A4, observe

[itex]A^3=(5A+2I_2)A=5A^2+2A=5(5A+2I_2)+2A=27A+10I_2[itex]
[itex]A^4=A^3A=(27A+10I_2)A=27A^2+10A=27(5A+2I_2)+10A[itex]
[itex]A^4=145A+54I_2.[itex]

The theorem is also an important tool in calculating eigenvectors.

General statement and proof

The proof of the Cayley-Hamilton theorem, even in its more general manifestations, is corollary to Cramer's rule from linear algebra.

In general, the Cayley-Hamilton theorem holds for endomorphisms of finitely generated modules over any commutative ring. Let M be a finitely generated module over a commutative ring R. If [itex]\phi:M\rightarrow M[itex] is an endomorphism of R-modules, the characteristic polynomial of [itex]\phi[itex] can be defined as follows. Let [itex]m_i,\ i=1,2,...,n[itex] be any collection of generators of M. In terms of these generators, [itex]\phi(m_i)=\sum a_{ij}m_j[itex] for some square matrix [itex]A=(a_{ij})[itex] with entries in R. It is now possible to define the characteristic polynomial of [itex]\phi[itex] by p(t)=det(A-tI), where the determinant is computed via Laplace expansion, permutations, partial Gaussian elimination (PLU-factorization), or any other convenient technique. As long as division does not occur, these yield the same result for the determinant (see Strang, Introduction to Linear Algebra or van der Waerden, Modern Algebra for details).

The Cayley-Hamilton theorem then states that p(A)=0. This more general version of the theorem is the source of the celebrated Nakayama lemma in commutative algebra and algebraic geometry.

Before commencing with the proof, we shall review Cramer's rule in its full generality. Let B be a commutative ring and S be an nxn matrix with entries in B. If Adj(S) denotes the matrix of cofactors of S, then Adj(S)S=det(S)I where I is the identity matrix.

Proof of Cayley-Hamilton theorem. Let R[t] be the ring of polynomials over R. The matrix A above provides a homomorphism of R[t] into the endomorphism ring of M by evaluating at t=A. (For example, if [itex]f(t)=3t^2+t+1[itex] is in the ring of polynomials over R, then the evaluation of f at t=A is [itex]f(A)=3A^2+A+I[itex].)

By Cramer's rule (over the commutative ring B=R[t]) we have Adj(A-tI)(A-tI)=det(A-tI)I=p(t)I. In particular, if [itex]m\in M[itex] is arbitrary, then Adj(A-tI)(Am-tm)=p(t)m. After an application of the evaluation homomorphism to both sides of the equation, we conclude that p(A)m=0 for every m, and therefore p(A)=0.ja:ケイリー・ハミルトンの定理

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