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 is a given matrix and is the identity matrix, then the characteristic polynomial of is defined as , where is the determinant operation and is a variable for a scalar element of the base ring. Since the entries of the matrix are (linear or constant) polynomials in , the determinant is also a degree- monic polynomial in , One can create an analogous polynomial in the matrix instead of the scalar variable , 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 to be expressed as a linear combination of the lower matrix powers of . 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 real or complex matrices. Cayley in 1858 stated the result for and smaller matrices, but only published a proof for the case. As for 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.
Examples
matrices
For a matrix , the characteristic polynomial is given by , and so is trivial.
matrices
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 matrix,
the characteristic polynomial is given by , so the Cayley–Hamilton theorem states that
which is indeed always the case, evident by working out the entries of .
Applications
Determinant and inverse matrix
For a general invertible matrix , i.e., one with nonzero determinant, −1 can thus be written as an order polynomial expression in : As indicated, the Cayley–Hamilton theorem amounts to the identity
The coefficients are given by the elementary symmetric polynomials of the eigenvalues of . Using Newton identities, the elementary symmetric polynomials can in turn be expressed in terms of power sum symmetric polynomials of the eigenvalues:
where is the trace of the matrix . Thus, we can express in terms of the trace of powers of .
In general, the formula for the coefficients is given in terms of complete exponential Bell polynomials as
In particular, the determinant of equals . Thus, the determinant can be written as the trace identity:
Likewise, the characteristic polynomial can be written as
and, by multiplying both sides by (note ), one is led to an expression for the inverse of as a trace identity,
Another method for obtaining these coefficients for a general matrix, provided no root be zero, relies on the following alternative expression for the determinant,
Hence, by virtue of the Mercator series,
where the exponential only needs be expanded to order , since is of order , the net negative powers of automatically vanishing by the C–H theorem. (Again, this requires a ring containing the rational numbers.) Differentiation of this expression with respect to allows one to express the coefficients of the characteristic polynomial for general as determinants of matrices,
Examples
For instance, the first few Bell polynomials are = 1, , , and .
Using these to specify the coefficients of the characteristic polynomial of a matrix yields
The coefficient gives the determinant of the matrix, minus its trace, while its inverse is given by
It is apparent from the general formula for cn−k, expressed in terms of Bell polynomials, that the expressions
always give the coefficients of and of in the characteristic polynomial of any matrix, respectively. So, for a matrix , the statement of the Cayley–Hamilton theorem can also be written as
where the right-hand side designates a matrix with all entries reduced to zero. Likewise, this determinant in the case, is now
This expression gives the negative of coefficient of in the general case, as seen below.
Similarly, one can write for a matrix ,
where, now, the determinant is ,