# Matrix polynomial

A matrix polynomial is a linear combination of the powers of a square matrix.

## Powers

Remember that the powers of a square matrix are obtained by multiplying by itself several times.

In particular, given a positive integer and a matrix , we have

A commonly adopted convention is that the -th power of is the identity matrix:

## Definition

We can now define matrix polynomials.

Definition Let be a matrix. Let be a non-negative integer. We say that is a polynomial in of degree if and only ifwhere is the identity matrix, are scalars, and .

Thus, is a matrix having the same dimension as , obtained as a linear combination of powers of .

The scalars are the so-called coefficients of the matrix polynomial.

In the above definition is assumed to be a non-negative integer. If for any matrix (i.e., the matrix polynomial is identically equal to the zero matrix), then we adopt the convention that the degree of is .

Example Let be a square matrix. Then,is a matrix polynomial in of degree .

## Difference with respect to ordinary polynomials

Note that a matrix polynomial as defined above is not an ordinary polynomial. In fact, in an ordinary polynomial, the elements being raised to powers and their coefficients are required to come from the same field. Instead, in a matrix polynomial, the coefficients come from a field (the so-called field of scalars) while the matrices being raised to powers come from a different set, which is not even a field. For example, in a field the multiplication operation must be commutative, but matrix multiplication is not commutative. Actually, the set of matrices, for fixed , is a ring, an algebraic structure satisfying a weaker set of axioms than that satisfied by a field.

Nonetheless, 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 associated matrix polynomialprovided that the entries of the matrix belong to the field .

It is common practice to switch back and forth between these two kinds of polynomials. For example, we often: 1) write a matrix polynomial; 2) derive its associated ordinary polynomial; 2) use the theory of ordinary polynomials to write the polynomial in a different form (e.g., we factorize it); 3) use the new form of the ordinary polynomial (e.g., its factorization) to rewrite the original matrix polynomial.

Example Define the matrix polynomialIts associated ordinary polynomial iswhich can be re-written asThen, we have

In theory, every time that we switch back and forth between the two kinds of polynomials, we should check whether the properties of ordinary polynomials that we are using hold also for matrix polynomials.

In practice, if we revise previous lectures on ordinary polynomials (e.g., polynomial division and greatest common divisors), we will realize that basically every definition, proof and proposition in those lectures is valid also for matrix polynomials. The reason is that, even if we lose some properties of fields when we deal with matrices, we do not need those properties to manipulate polynomials. Moreover, we can even use the commutative property of products, as explained in the next section.

## Commutative property

We have already said that, in general, matrix multiplication is not commutative. However, the multiplication of two polynomials and in the same matrix is commutative:

This perhaps obvious fact (a proof of which can be found in a solved exercise at the end of this lecture) is extremely useful, and it is used to prove important results in linear algebra (e.g., the result below on null spaces).

## Polynomials in linear operators

We can also define polynomials in linear operators, but, as we will shortly argue, we do not need to develop a separate theory because (as long as we deal with finite-dimensional vector spaces) a polynomial in a linear operator is always associated to an analogous matrix polynomial and we can study the properties of the latter.

Definition Let be a vector space and a linear operator. Let be a non-negative integer. We say that is a polynomial in of degree if and only ifwhere are scalars, , is the identity operator that associates each member of to itself, and denotes the operator obtained by composing times :

Let be a basis for the vector space . Remember that, provided is finite-dimensional, any linear operator is associated to a square matrix, called matrix of the linear operator with respect to and denoted by such that, for any ,where and respectively denote the coordinate vectors of and with respect to .

Moreover, the composition of operators can be performed by multiplying their respective matrices:

Therefore, a polynomial such as the one defined above is an operator whose matrix is a matrix polynomial in .

Thus, as we have said at the beginning of this section, we do not need a separate theory for operator polynomials and we can deal with them by using polynomials in the respective matrices.

## Null space of a matrix polynomial

A simple albeit often used result follows.

Proposition Let be the space of vectors. Let be a matrix. Let be a polynomial in . Then, the null space of is an invariant subspace of under the linear transformation defined by .

Proof

Denote by the null space of . Choose any . We have Thus, , which proves that is invariant under .

## Eigenvalues of a matrix polynomial

Once we know the eigenvalues of , we can easily compute the eigenvalues of .

Proposition Let be a matrix. Let be a polynomial in . Let be an eigenvalue of . Then, is an eigenvalue of .

Proof

We have previously demonstrated that, if is an eigenvalue of associated to the eigenvector , then is an eigenvalue of associated to the same eigenvector . Suppose that Choose any eigenvalue of and an associated eigenvector . Then,which proves the proposition.

## Annihilating polynomial

A frequently used concept is that of an annihilating polynomial.

Definition Let be a matrix. Let be a polynomial in . We say that is an annihilating polynomial if and only if

## Cayley-Hamilton theorem

In the lecture on the Cayley-Hamilton theorem, we will carefully explain one of the most important results in the theory of matrix polynomials, which states that the characteristic polynomial is an annihilating polynomial.

## Solved exercises

Below you can find some exercises with explained solutions.

### Exercise 1

Let be a square matrix. Transform the ordinary polynomialinto a polynomial in .

Solution

The matrix polynomial is

### Exercise 2

Let be a square matrix. Transform the ordinary polynomialinto a polynomial in .

Solution

The matrix polynomial is

### Exercise 3

Prove that the multiplication of two polynomials and in the same matrix is commutative:

Solution

Suppose thatandThen