The Cayley-Hamilton theorem shows that the characteristic polynomial of a square matrix is identically equal to zero when it is transformed into a polynomial in the matrix itself. In other words, a square matrix satisfies its own characteristic equation.

In the lecture on matrix polynomials we have explained that, if is a field, such as the set of real numbers or the set of complex numbers , and is an ordinary polynomialthen we can use to define, by extension, an analogous matrix polynomialprovided that the entries of the square matrix belong to the field .

Remember that the characteristic polynomial of a square matrix iswhere are the eigenvalues of .

As explained above, an ordinary polynomial can be used to define a matrix polynomial .

The proposition in the next section, known as Cayley-Hamilton theorem, shows that the characteristic polynomial of is identically equal to zero when it is transformed into a polynomial in .

Here is the Cayley-Hamilton theorem.

Proposition Let be a matrix. Let be the eigenvalues of . Then,

Proof

DefineThe matrix has a Schur decompositionwhere is an upper triangular matrix, is a unitary matrix and denotes the conjugate transpose of . Moreover, the diagonal entries of are the eigenvalues of . Since is unitary, . Therefore,We are going to show that and, as a consequence, . DefineWe are going to prove by induction that the first columns of are zero (hence all the columns of are zero and the proposition is true). Let us start from . Since is upper triangular, is the only non-zero entry of the first column of . Therefore, the first column of the matrix is zero. Now suppose that the first columns of are zero, that is,The columns of can be seen as linear combinations of the columns of with coefficients taken from the corresponding columns of . In particular, the -th column of is where: in step we have used the fact that is upper triangular and, as a consequence, when ; in step we have used the fact that the first columns of are zero. When is one of the first columns of , then . If , then the summation is over an empty set of indices andIf , then the summation is over a single index andbecause Thus, we have reached the desired conclusion: the first columns of are zero. It follows that all the columns of are zero and .

Thus, if is the characteristic polynomial of , then

Let us make an example.

Example DefineThe characteristic polynomial of isWe can transform it into a polynomial in as follows:Let us carry out the multiplications so as to check that indeed the Cayley-Hamilton theorem holds:

An important consequence of the Cayley-Hamilton theorem is that any polynomial in a matrix can be rewritten as a polynomial whose degree is at most .

Proposition Let be a matrix. Let be a matrix polynomial in . Then, there exists a polynomial such thatand the degree of is at most .

Proof

If the degree of is less than , then there is nothing to prove. If the degree of is greater than or equal to , we proceed as follows. By the Cayley-Hamilton theorem, we havewhere the scalars are obtained by expanding the product . Thus, can be expressed as a linear combination of powers of up to the -th:If we pre-multiply both sides of the previous equation by , we obtainBy substituting (1) into (2), we obtain that also can be expressed as a linear combination of powers of up to the -th. With this technique (pre-multiply and substitute), we obtain the same result for any power , where is any positive integer. Thus, given a polynomial of degree greater than or equal to , we can substitute all the powers of that appear in the polynomials and are greater than or equal to with lower powers. Hence, we have the stated result.

Below you can find some exercises with explained solutions.

DefineRe-write the polynomialas a polynomial of degree in .

Solution

The characteristic polynomial of isTherefore, by the Cayley-Hamilton theorem, we haveWe pre-multiply both sides of the previous equation so as to obtainThen, we substitute (3) into (4) and getFinally, we can re-write the given polynomial

Please cite as:

Taboga, Marco (2021). "Cayley-Hamilton theorem", Lectures on matrix algebra. https://www.statlect.com/matrix-algebra/Cayley-Hamilton-theorem.

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