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Concept
Cayley–Hamilton theorem
Summary
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.
If A is a given n × n matrix and In is the n × n identity matrix, then the characteristic polynomial of A is defined as , where det is the determinant operation and λ is a variable for a scalar element of the base ring. Since the entries of the matrix are (linear or constant) polynomials in λ, the determinant is also a degree-n monic polynomial in λ, One can create an analogous polynomial in the matrix A instead of the scalar variable λ, defined as The Cayley–Hamilton theorem states that this polynomial expression is equal to the zero matrix, which is to say that . The theorem allows An to be expressed as a linear combination of the lower matrix powers of A. When the ring is a field, the Cayley–Hamilton theorem is equivalent to the statement that the minimal polynomial of a square matrix divides its characteristic polynomial.
A special case of the theorem was first proved by Hamilton in 1853 in terms of inverses of linear functions of quaternions. This corresponds to the special case of certain 4 × 4 real or 2 × 2 complex matrices. Cayley in 1858 stated the result for 3 × 3 and smaller matrices, but only published a proof for the 2 × 2 case. 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”. The general case was first proved by Ferdinand Frobenius in 1878.
For a 1 × 1 matrix A = (a), the characteristic polynomial is given by p(λ) = λ − a, and so p(A) = (a) − a(1) = 0 is trivial.
As a concrete example, let
Its characteristic polynomial is given by
The Cayley–Hamilton theorem claims that, if we define
then
We can verify by computation that indeed,
For a generic 2 × 2 matrix,
the characteristic polynomial is given by p(λ) = λ2 − (a + d)λ + (ad − bc), so the Cayley–Hamilton theorem states that
which is indeed always the case, evident by working out the entries of A2.
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