Cayley–Hamilton theorem

In linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Rowan Hamilton) states that every square matrix over a commutative ring (such as the real or complex numbers or the integers) satisfies its own characteristic equation.

, where det is the determinant operation, λ is a variable scalar element of the base ring, and In is the n × n identity matrix.

A special case of the theorem was first proved by Hamilton in 1853[6] in terms of inverses of linear functions of quaternions.

[7][8] As for n × n matrices, Cayley stated “..., I have not thought it necessary to undertake the labor of a formal proof of the theorem in the general case of a matrix of any degree”.

For a general n × n invertible matrix A, i.e., one with nonzero determinant, A−1 can thus be written as an (n − 1)-th order polynomial expression in A: As indicated, the Cayley–Hamilton theorem amounts to the identity

In general, the formula for the coefficients ci is given in terms of complete exponential Bell polynomials as[nb 1]

Another method for obtaining these coefficients ck for a general n × n matrix, provided no root be zero, relies on the following alternative expression for the determinant,

The increasingly complex expressions for the coefficients ck is deducible from Newton's identities or the Faddeev–LeVerrier algorithm.

None of these computations, however, can show why the Cayley–Hamilton theorem should be valid for matrices of all possible sizes n, so a uniform proof for all n is needed.

This holds for all possible eigenvalues λ, so the two matrices equated by the theorem certainly give the same (null) result when applied to any eigenvector.

While this provides a valid proof, the argument is not very satisfactory, since the identities represented by the theorem do not in any way depend on the nature of the matrix (diagonalizable or not), nor on the kind of entries allowed (for matrices with real entries the diagonalizable ones do not form a dense set, and it seems strange one would have to consider complex matrices to see that the Cayley–Hamilton theorem holds for them).

We shall therefore now consider only arguments that prove the theorem directly for any matrix using algebraic manipulations only; these also have the benefit of working for matrices with entries in any commutative ring.

The simplest proofs use just those notions needed to formulate the theorem (matrices, polynomials with numeric entries, determinants), but involve technical computations that render somewhat mysterious the fact that they lead precisely to the correct conclusion.

It is possible to avoid such details, but at the price of involving more subtle algebraic notions: polynomials with coefficients in a non-commutative ring, or matrices with unusual kinds of entries.

This proof uses just the kind of objects needed to formulate the Cayley–Hamilton theorem: matrices with polynomials as entries.

While this looks like a polynomial with matrices as coefficients, we shall not consider such a notion; it is just a way to write a matrix with polynomial entries as a linear combination of n constant matrices, and the coefficient t i has been written to the left of the matrix to stress this point of view.

So when considering polynomials in t with matrix coefficients, the variable t must not be thought of as an "unknown", but as a formal symbol that is to be manipulated according to given rules; in particular one cannot just set t to a specific value.

respecting the order of the coefficient matrices from the two operands; obviously this gives a non-commutative multiplication.

In the first proof, one was able to determine the coefficients Bi of B based on the right-hand fundamental relation for the adjugate only.

Indeed, even over a non-commutative ring, Euclidean division by a monic polynomial P is defined, and always produces a unique quotient and remainder with the same degree condition as in the commutative case, provided it is specified at which side one wishes P to be a factor (here that is to the left).

In addition to proving the theorem, the above argument tells us that the coefficients Bi of B are polynomials in A, while from the second proof we only knew that they lie in the centralizer Z of A; in general Z is a larger subring than R[A], and not necessarily commutative.

Since A is an arbitrary square matrix, this proves that adj(A) can always be expressed as a polynomial in A (with coefficients that depend on A).

Note that this identity also implies the statement of the Cayley–Hamilton theorem: one may move adj(−A) to the right hand side, multiply the resulting equation (on the left or on the right) by A, and use the fact that

It is clearer to distinguish A from the endomorphism φ of an n-dimensional vector space V (or free R-module if R is not a field) defined by it in a basis

2.4) (which in fact is the more general statement related to the Nakayama lemma; one takes for the ideal in that proposition the whole ring R).

is the element whose component i is ei (in other words it is the basis e1, ..., en of V written as a column of vectors).

the associativity of matrix-matrix and matrix-vector multiplication used in the first step is a purely formal property of those operations, independent of the nature of the entries.

for some sequence of elements e1, ..., en that generate V (which space might have smaller dimension than n, or in case the ring R is not a field it might not be a free module at all).

of some B-module M (supposed to be free and of finite rank) have been used by Gatto & Salehyan (2016, §4) to prove the Cayley–Hamilton theorem.

A proof based on developing the Leibniz formula for the characteristic polynomial was given by Straubing[18] and a generalization was given using trace monoid theory of Foata and Cartier.